Diplomarbeit

Analyticity properties of the decompositionof the spectral density matrix in the context

of dynamic PCA

ausgeführt am Institut für Wirtschaftsmathematik derTechnischen Universität Wien

unter der Anleitung vonO. Univ. Prof. Dipl.-Ing. Dr. tech. Manfred Deistler

durch

Bernd FunovitsMatrikelnummer 0451617G.A. Wimmerplatz 1/2/7

7432 Oberschützen

Oberschützen am 10. August 2009 ____________________

Danksagung

“Der Realist ist insofern naiv, als er nicht zur Kenntnis nimmt, daß

wir nicht in der Welt leben, sondern in dem Bild, das wir uns von

der Welt machen.”

Hoimar von Ditfurth

Mein Bild, das ich mir von der Welt gemacht habe, wurde durch Prof. Manfred

Deistlers offene und unvoreingenommene Herangehensweise an Problemstellun-

gen jeglicher Art wesentlich beeinflusst. Dafür möchte ich mich auf diesem Wege

ganz herzlich bedanken. Seine Vorlesungen über Ökonometrie und Zeitreihen

sowie das zugehörige Seminar haben für mich einen beachtlichen Mehrwert bei

meinem zweijährigen Studium Aufenthalt in Frankreich geschaffen und wer-

den mir sicherlich auch weiterhin sehr zugute kommen. Weiters schätzte ich

ganz besonders, dass der Kontakt und die persönliche Betreuung auch während

meiner Abwesenheit von der TU Wien niemals abriss und dass Prof. Deistler

mir die Möglichkeit gab an Vortragsreihen in Brüssel und London anwesend zu

sein. Sein disziplinenübergreifendes Interesse für Mathematik hat letztendlich

auch zu dem Thema dieser Diplomarbeit geführt.

Ganz besonderer Dank gebührt meiner Familie die mir zu jeder Zeit Verständnis

und Unterstützung für die Art und Weise wie ich zu studieren pflegte entgegen-

brachte.

1

Contents

1 Principal component analysis in the frequency domain 4

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Description in mathematical terms . . . . . . . . . . . . . 4

1.1.2 Static and quasi-static case . . . . . . . . . . . . . . . . . 5

1.1.3 Definition of principal component analysis . . . . . . . . . 5

1.1.4 Example: Impulse series of earthquakes . . . . . . . . . . 6

1.1.5 Choice of r . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Static case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Dynamic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Decomposition of the spectral density . . . . . . . . . . . . . . . 13

2 Normed commutative algebras 15

2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Equivalence classes with respect to ideals . . . . . . . . . . . . . 20

2.4 The canonical isomorphism on the maximal ideals . . . . . . . . 23

2.4.1 The relation between maximal ideals and complex multi-

plicative homomorphisms . . . . . . . . . . . . . . . . . . 24

2.5 Proof of the univariate case . . . . . . . . . . . . . . . . . . . . . 27

2.6 The topology onM(R) . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Generating systems and polynomially convex sets . . . . . . . . . 31

2.8 Proof of the multivariate case . . . . . . . . . . . . . . . . . . . . 35

2.8.1 Case A: Algebra R is generated by a finite number of

elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.8.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . 39

2

3 Perturbation of Polynomials 42

3.1 Polynomials with Meromorphic Coefficients . . . . . . . . . . . . 46

3.2 Behavior of the roots near critical points . . . . . . . . . . . . . . 52

3.3 Continuous representation of the root functions . . . . . . . . . . 58

4 Smoothness of Eigenprojections and Eigenvectors 61

4.1 Some facts about resolvents and eigenprojections of matrices . . 61

4.2 Integral representation of the eigenprojections . . . . . . . . . . . 64

4.3 Analyticity at non-critical points . . . . . . . . . . . . . . . . . . 66

4.3.1 Analyticity with respect to matrices . . . . . . . . . . . . 67

4.3.2 Analyticity with respect to a parameter . . . . . . . . . . 68

4.3.3 Analyticity and Construction of the generalized eigenbasis 72

4.4 Analytic properties of eigenprojections near critical points . . . . 74

4.5 Real Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Bibliography 86

3

1 Principal component analysis in the frequency

domain

1.1 Introduction

The analysis of multivariate time series is an important issue in many research

areas, such as economics, finance, signal processing and medicine. Multivariate

time series are modeled jointly when the relation between the single time series

or co-movements are important. In general, the number of free parameters,

which is a measure of the complexity of the model class considered, increases

substantially with the number of observed variables. E.g, if the cross-sectional

dimension in a VAR(p) model is n, the dimension of the parameter space is

proportional to n2 (the so-called curse of dimensionality). Thus, the complexity

of the model class shows quadratic dependence on n, whereas the number of

data points, for fixed sample size T is linear in n. Factor models, in particular

the PCA model, mitigate this problem.

1.1.1 Description in mathematical terms

The basic common equation for all different kinds of factor models considered

here is of the form

xt = Λ(z)ξt + ut

= χt + utt ∈ Z,

where xt is the n-dimensional vector of observations, ξt is the r-dimensional

factor, ut is the n-dimensional noise and Λ(z) =∑∞j=−∞ Λjzj, Λj ∈ Rn×r

denotes the transfer function. Thereby, the argument z is used as complex

variable as well as back-shift operator on Z. χt = Λ(z)ξt is called the common

component or the latent variable.

Throughout the first chapter the following is assumed:

• E(ξt) = E(ut) = 0, ∀t ∈ Z

• E(ξtusT ) = 0 ∈ Rr×n, ∀t, s ∈ Z

• (ξt)t∈Z, (ut)t∈Z are wide sense stationary and (linearly) regular

4

• The covariances γξ(s) = E(ξtξt+sT ) ∈ Rr×r and γu(s) = E(utut+sT ) ∈

Rn×n are satisfying

∞∑

s=−∞

‖γξ(s)‖ <∞,∞∑

s=−∞

‖γu(s)‖ <∞,

where ‖·‖ denotes an arbitrary matrix norm.

•∞∑

j=−∞

‖Λj‖ <∞

• The spectral density fχ(λ) of (χt)t∈Z has rank r for all λ ∈ [−π, π].

Due to the summability condition of γξ(s), γu(s) and Λ(z) and due to the fact

that the processes ξt and ut are uncorrelated, the spectral densities fξ of (ξt)t∈Z

and fu of (ut)t∈Z exist as uniform limits of trigonometric polynomials and the

spectral density fx of the observation process (xt)t∈Z can be represented as

fx(λ) = Λ(e−iλ)fξ(λ)Λ(e−iλ)T

+ fu(λ).

1.1.2 Static and quasi-static case

When Λ(z) = Λ is constant and (ξt)t∈Z, (ut)t∈Z and thus (xt)t∈Z are white

noises, the factor model is called static.

The variance matrix of (xt)t∈Z, Σx ∈ Rn×n, is decomposed as

Σx = ΛΣξΛT + Σu.

When (ξt)t∈Z, (ut)t∈Z are not necessarily white noises, the model is called quasi-

static.

For given fx, or Σx respectively, too many models would be possible, see [14].

Thus, in order to obtain reasonable model classes, further assumptions have to

be imposed. This leads to principal component models, linear factor models

with idiosyncratic noise and generalized linear factor model. In this work, only

the first-mentioned will be considered.

1.1.3 Definition of principal component analysis

The aim of principal component analysis (PCA) is to approximate the n-

dimensional observed process (xt)t∈Z by a filtered version of itself whose spectral

5

density is of reduced rank r such that the variance of the residuals is minimized.

Hence, the additional assumption in the PCA-model is, that ξt = C(z)xt holds.

The (r × n)-dimensional filter C(z) and the (n× r)-dimensional filter Λ(z) are

obtained by minimizing for fixed reduced rank r

trE(utut

T )

= tr

E[(xt − Λ(z)C(z)xt)(xt − Λ(z)C(z)xt)T ],

where tr denotes the trace.

1.1.4 Example: Impulse series of earthquakes

A situation in which such a model of reduced rank might be employed is the

following: Let ξt represent the impulse series of r earthquakes occurring si-

multaneously at various locations; let xt represent the signals received by n

seismometers; let Λ(z) represent the transmission effects of the earth on the

earthquakes. Seismologists are interested in investigating the series (ξt)t∈Z.

1.1.5 Choice of r

For the PCA - model, the number of factors r is not intrinsic in the sense, that

it is not a property of fx or Σx. By the choice of r, the degree of dimension

reduction and, as a trade-off, the quality of approximation are determined.

1.2 Static case

As already mentioned above, in the static case of the principal component anal-

ysis Λ(z) = Λ ∈ Rn×r and C(z) = C ∈ Rr×n are constant.

The solution of the minimization problem

minΛ∈R

n×r

C∈Rr×n

tr

E[(xt − ΛCxt)(xt − ΛCxt)T ],

is obtained by the eigenvalue decomposition of the variance-covariance matrix

of the random variable xt.

The following theorem treats the case where E(x) = c 6= 0 ∈ Rn.

Theorem 1.1 (Solution of the static minimization problem).

Let xt be a n-dimensional vector-valued random variable with the properties

6

• E(xt) = cx ∈ Rn, ∀t ∈ Z

• E[(xt − cx)(xt − cx)T ] = Σx.

Then, the solution of the minimization problem

minΛ∈R

n×r

C∈Rr×n

d∈Rn

tr

E[(xt − (d+ ΛCxt))(xt − (d+ ΛCxt))T ]

is given by

• d = cx − ΛCcx

• C =

. . . o1T . . .

. . . o2T . . .

. . .... . . .

. . . orT . . .

=: O1T , Λ =

......

......

o1 o2 . . . or...

......

...

= O1,

where the oj are the eigenvectors of Σx.

The obtained minimal value is

minΛ∈R

n×r

C∈Rr×n

d∈Rn

tr

E((xt − (d+ ΛCxt)) (xt − (d+ ΛCxt))T )

=∑

j>r

µj ,

where the µj are the eigenvalues of Σx and µj+1 ≤ µj for all j ∈ 1, . . . , n− 1

.

Proof. See [2].

Post-multiplying Λ with a non-singular matrix P ∈ Rn×n and pre-multiplying

ξt with its inverse P−1 yields the same χt. However, by making the special

choice

ξt = O1Txt, Λ = O1, ut = O2O2

Txt

the matrices in the decomposition of Σx

Σx = O1Ω1O1T +O2Ω2O2

T =

= Σχ + Σu,

can be uniquely identified. In this representation, Ω1 ∈ Rr×r and Ω2 ∈

R(n−r)×(n−r) denote the diagonal matrices consisting of the r largest and (n−r)

7

smallest eigenvalues of Σx arranged in decreasing order and O1 = (o1 · · · or) and

O2 = (or+1 · · · on) the (n × r)- and [n × (n − r)]-dimensional matrices of the

corresponding eigenvectors.

Definition 1.2 (Principal component ξt of xt). The variate ξjt = ojTxt, j ∈

1, . . . , n, t ∈ Z is called the j-th principal component of (xt)t∈Z.

• The principal components can be interpreted as the coordinates of xt in

the eigenbasis of Σx.

If xt follows a normal distribution with n dimensions, i.e. Nn(0,Σx), the com-

ponents ξjt of the vector of principal components are independent and N1(0, µj)-

distributed, where µj denotes the j-th largest eigenvalue of Σx.

Let Xm,m ∈ 1, . . . , T be a sample from Nn(0,Σx) and define X =

[X1 · · ·XT ] ∈ Rn×T . An estimator for Σx is

Σx =XXT

T.

The j-th largest eigenvalue µj of Σx is estimated by the j-th largest eigenvalue

µj of Σx and the corresponding eigenvector oj by oj .

Theorem 1.3 (Results for estimators). Under the assumption that

• X1, . . . , XT is a sample from Nn(0,Σx) and that

• the eigenvalues of Σx are distinct,

it follows, that

• the members of the set µj , oj : j ∈ 1, . . . , n are asymptotically normal

and

• µj, : j ∈ 1, . . . , n and oj : j ∈ 1, . . . , n are asymptotically inde-

pendent.

Furthermore, the asymptotic moments are given by

• E(µj) = µj +O(T−1)

8

• E(oj) = oj +O(T−1)

• Cov(µj , µk) =

δjkµ2j

T +O(T−2) j = k

O(T−2) j 6= k

• Cov(oj , ok) =

µj∑l 6=j

µl(µj−µl)2

ololT

T +O(T−2) j = k

O(T−2) j 6= k

• Cov(oj , ok) =

− µkµj

(µj−µl)2

okoTj

T +O(T−2) j = k

O(T−2) j 6= k

where j, k ∈ 1, . . . , n.

Proof. See [2].

The theorem above results mainly from two facts:

• The indicated eigenvalues and vectors are differentiable functions of the

entries of Σx.

• Σx is asymptotically normal as T →∞.

The matrix Σx has a Wishart distribution, which is a multidimensional gener-

alization of the χ2-distribution, and the distribution of µj can be approximated

by µjχ22T

2T .

1.3 Dynamic case

For the dynamic generalization of the PCA, the spectral density fx and its

eigenvalue decomposition are considered in a similar way. The filter O1(z) of

reduced rank which provides the best approximation of the process by itself will

be deduced.

The basic equation is again

xt = O1(z)ξt + ut = χt + ut, t ∈ Z.

This matrix function can, due to the fact that (ut)t∈Z and (ξt)t∈Z are uncorre-

lated, be decomposed as

fx(λ) = O1(e−iλ)Λ1(λ)O1(e−iλ)T

+ O2(e−iλ)Λ2(λ)O2(e−iλ)T

, (1)

9

where O1(eiλ) and O2(eiλ) are the corresponding transfer function to the (n×r)-

and [n× (n− r)]-dimensional filters O1(z) and O2(z). For the sake of simplicity,

the transfer functions are, from now on, denoted as O1(λ) and O2(λ).

Theorem 1.4 (Solution of the minimization problem). Let (xt)t∈Z be a n-

dimensional wide sense stationary stochastic process with absolutely summable

covariance function satisfying the conditions

• E(xt) = cx ∀t ∈ Z,

• Cov(xt+u, xt) = γx(u) ∈ Rn×n, u ∈ Z is absolutely summable and

• fx(λ) = 12π

∞∑u=−∞

γx(u)e−iuλ ∈ Cr×r is positive-definite for every λ ∈ R.

The solution of the minimization problem

minΛ(z)C(z)d∈R

n

tr

E((xt − (d+ Λ(z)C(z)xt)) (xt − (d+ Λ(z)C(z)xt))T ), (2)

where

• Λ(z) =∞∑

j=−∞

Λjzj,Λj ∈ Rn×r is a (n× r)-dimensional filter and

• C(z) =∞∑

j=−∞

Cjzj , Cj ∈ Rr×n is a (r × n)-dimensional filter,

is given by

• d = cx − Λ(z)C(z)cx

• C(λ) =

. . . o1(λ)T

. . .

. . . o2(λ)T

. . .

. . .... . . .

. . . or(λ)T

. . .

,Λ(λ) =

......

...

o1(λ) . . . or(λ)...

......

=

CT

(λ)

• Λj =2π∫0

Λ(λ)eijαdα, Cj =2π∫0

C(λ)eijαdα,

where the oj(λ) are the eigenvectors of fx(λ).

The obtained minimal value is∫ 2π

0

∑

j>r

µj(α)dα,

10

where µj(λ) is the j-th eigenvalues of fx(λ).

Proof. Step 1: Reformulation of the minimization problem.

Since

• E(Y T · Y ) = E(Y − E(Y ))T · (Y − E(Y )) + E(Y )TE(Y ) and

• trE(Y Y T )

= E(tr

Y Y T

) = E(tr

Y TY

) = tr

E(Y TY )

=

E(Y TY ),

where Y is a n-dimensional random variable and by defining

εt := xt − (d+ Λ(z)C(z)xt)− E(xt − (d+ Λ(z)C(z)xt))

= xt − (d+ Λ(z)C(z)xt)− cx + d+ Λ(z)C(z)cx,

the expression

tr

E((xt − (d+ Λ(z)C(z)xt)) (xt − (d+ Λ(z)C(z)xt))T )

is minimal if and only if

E[(xt − d− Λ(z)C(z)xt)T ]E[(xt − d− Λ(z)C(z)xt)] + E(εt

T εt) (3)

is minimal.

Step 2a: Minimization of the first term.

The first term in (3) is annihilated by setting

d := cx − Λ(z)C(z)cx.

Step 2b: Minimization of the second term.

The second term in (3) can be written as

E(εtT εt) = E(trεtεt

T)

=

= trE((εtεtT ))

=

= tr γε(0)

= tr∫ 2π

0 ei0λfε(λ)dλ

By defining A(λ) := Λ(eiλ)C(eiλ)

fε(λ) = (I −A(λ))fx(λ)(I −A(λ))T.

11

Since fx(λ) is positive-definite fx(λ)12 = O(λ) ·

√Ω(λ) · O(λ)

Texists, fx(λ) =

fx(λ)T

and therefore

tr[(I −A(λ))fx(λ)(I −A(λ))

T]

=

= tr

[(fx(λ)

12 −A(λ)fx(λ)

12 )(fx(λ)

12 −A(λ)fx(λ)

12 )T]

is minimized over all transfer function A(λ) with rank smaller or equal to r by

A(λ) =r∑

j=1

oj(λ)oj(λ)T.

The minimal value of (2) is

tr

(O(λ)

√Ω(λ)O(λ)

T−A(λ)O(λ)

√Ω(λ)O(λ)

T)︸ ︷︷ ︸

=∗

(∗)T

=

= tr

(O(λ)√

Ω(λ)O(λ)T−r∑

j=1

oj(λ)oj(λ)TO(λ)

√Ω(λ)O(λ)

T)

︸ ︷︷ ︸=∗

(∗)T

=

tr(Ωn−r(λ)),

where Ωn−r(λ) denotes the (n− r) × (n− r) diagonal matrix containing those

(n− r) smallest eigenvalues

12

1.4 Decomposition of the spectral density

By considering the expression (1) on page 9, many interesting questions arise.

• Do all the eigenvalues of fx(λ) have to be distinct that they can repre-

sented as analytic or continuous functions?

If they are distinct, it follows from the implicit functions theorem that

they can be represented, under certain assumptions, as differentiable

functions. However, the question will be answered positively by a

theorem of Kato [11].

• If the coefficients of fx(λ) satisfy a summability condition which permit to

state that stationary input creates stationary output, is this summability

condition also fulfilled by the transfer function O1(λ)T

?

Every element in the transfer function O1(λ)T

is an analytical transformation

of the coefficients of the spectral density. Given that the coefficients of the

spectral density satisfy a summability condition (which will be stated in the

following theorem), it has to be shown that the analytical transformation of the

coefficients of fx(λ) satisfy the same summability condition in order to prove

the aforementioned question positively.

The miscellaneous summability conditions on the coefficients of the spectral

density are stated by the means of the normed commutative Banach algebra

V (l) =

z(λ) =

∞∑

u=−∞

a(u)e−iuλ, λ ∈ R | a(u) ∈ C ∀u ∈ Z ∧

∞∑

u=−∞

(1 + |u|l)|a(u)| <∞

.

Theorem 1.5 (Transformation of multivariate time series). Under the assump-

tions

• that the elements f(i,j)x (λ) of fx(λ) are elements of V (p), i.e.

∞∑

u=−∞

(1 +

|u|p)|γ(i,j)x (u)| < ∞ ∀i, j ∈ 1, . . . , n, where γ

(i,j)x (u) denotes the (i, j)-

element of the covariance function of (xt)t∈Z,

• that fx(λ) is self-adjoint, i.e. fx(λ) = fx(λ)T∀λ ∈ (−π, π], and l is its

maximum number of distinct eigenvalues and

13

• that fx(λ) can be continued analytically in C across a neighborhood which

contains the set(f (i,j)(λ)

)ni,j=1

| λ ∈ (−π, π]

and that it is there self-

adjoint as well.

it follows that

• the transfer functions O1(λ) and O1(λ)T

are in V (p) as well, i.e.

for every coefficient O(i,j)1 (λ) of O1(λ) exists a sequence

(b(i,j)(u))u∈Z, such that

O(i,j)1 (λ) =

∑∞u=−∞ b(i,j)(u)e−iuλ

and∑∞u=−∞(1 + |u|p)|b(i,j)(u)| <∞

and

for every coefficient O(i,j)1 (λ)

T

of O1(λ)T

exists a sequence

(c(i,j)(u))u∈Z, such that

O(i,j)1 (λ)

T

=∞∑

u=−∞c(i,j)(u)e−iuλ

and∞∑

u=−∞(1 + |u|p)|c(i,j)(u)| <∞.

Furthermore, the spectral density matrices fξ(λ) and fε of (ξ)t∈Z and (εt)t∈Z

respectively are in V (p) too.

Proof. By corollary 4.21, the coefficients of the transfer functions O1(λ) and

O1(λ)T

are analytic functions of the entries of fx(λ).

Therefore, by lemma 2.37, the summability condition is satisfied by the entries

of O1(λ) and O1(λ)T

as well.

Furthermore, the spectral density matrices fξ(λ) and fε of (ξ)t∈Z and (εt)t∈Z

respectively are in V (p) too, because ξt = O1(z)xt and εt = O2(z)xt.

14

2 Normed commutative algebras

In this chapter, it will be proved that the analytic transformation of several

elements of an commutative normed algebra is an element of the algebra as well.

This result, corollary 2.1, is then applied to the elements of the spectral density

matrix fx(λ) which are contained within V (l) of a stochastic process (xt)t∈Z

with the properties stated in the preceding chapter. The elements of fx(λ)’s

eigenbasis are as analytic transformation of elements of V (l) again contained

within V (l). The fact that the eigenbasis depends analytically on the entries of

fx(λ) and therefore on λ ∈ (−π, π] will be proved in the last chapter.

Firstly, the result will be proved for the case, where only one member of the

algebra is transformed analytically. Secondly, the theorem will be proved for the

multidimensional case. Except for some explicitly emphasized facts, the whole

section is based on [6].

As already mentioned above, the following corollary is the main result of this

chapter. It follows directly from theorem 2.37.

Corollary 2.1. If zj(λ) belongs to V (l), j ∈ 1, . . . , n, and

f(ζ1, . . . , ζn) is analytic on a neighborhood of the range of values

(z1(λ), . . . , zn(λ)) | λ ∈ (−π, π], then f(z1(λ), . . . , zn(λ)) also belongs to V (l).

In order to prove the univariate case, some theory about maximal ideals, their

relation to complex multiplicative homomorphisms on the algebra and their

quotient algebras must be developed.

2.1 Fundamentals

Definition 2.2 (Normed commutative algebra R). A normed commutative al-

gebra R is a complex Banach space with an associative, commutative and left-

and right-continuous multiplication

(·, ·) :

R×R → R

(x, y) 7→ x · y.

For every normed algebra without neutral element e for multiplication, a formal

neutral element can be adjoined. One defines the algebra of the formal sums

15

λe + x, λ ∈ C and x ∈ R. Therefore, it is assumed without loss of generality

that the normed commutative algebra have a neutral element for multiplication.

Theorem 2.3 (Norm-inequality). For every normed algebra R with neutral

element e 6= 0, there exists a norm on R which induces the same topology as the

initial one and which satisfies

‖x · y‖ ≤ ‖x‖‖y‖ ∀ x, y ∈ R and (4)

‖e‖ = 1. (5)

Proof. See [13], theorem 10.2.

Because of the preceding theorem, it is assumed that the norm ‖·‖ of the algebra

R satisfies the condition

‖x · y‖ ≤ ‖x‖‖y‖, ∀x, y ∈ R.

Definition 2.4 (Absolutely convergent series). All the xi are contained within

the algebra R. The series limN→∞∑Ni=1 xi is called absolutely convergent if and

only if the series

limN→∞

N∑

i=1

‖xi‖

converges.

A neighborhood of an element x ∈ R with radius ε is denoted by

Uε(x) := y ∈ R | ‖x− y‖ < ε

The collection of all the neighborhoods of x ∈ R is called the neighborhood

system. The neighborhood system is denoted by U(x).

Furthermore, a superscript indicates the underlying space, if it is not evident.

2.2 Maximal ideals

Theorem 2.5 (The set of inverse elements is open).

• O =x ∈ R | ∃ x−1

is an open set in the norm-topology.

16

• (·)−1 :

O → R

x 7→ x−1is a continous map from O on to R.

Proof. Step 1a: If the distance between x ∈ R and the neutral el-

ement e is smaller than 1, the inverse element exists and can be

represented as series, i.e. ‖e− x‖ < 1 =⇒ ∃x−1,∑∞n=0(e− x)n = x−1

The series e+ (e−x) + (e−x)2 + · · · converges absolutely because ‖(e−x)n‖ ≤

‖e− x‖n︸ ︷︷ ︸<1

. Since R is complete, this sum is also an element of R.

Multiplying this series with x = e− (e− x) leads to the telescoping sum

∞∑

n=0

(e−x)n ·(e− (e− x))︸ ︷︷ ︸=x

= e+(e−x)+(e−x)2+ · · ·−(e−x)−(e−x)2−· · · = e.

Thus, the series is the inverse element of x.

Step 1b: O is open, i.e. ∀x ∈ O =⇒ ∃U(x) ∈ U(x) : U(x) ⊆ O.

Due to the paragraph above, U1(e) = y ∈ R : ‖e− y‖ < 1 is a subset of O. It

follows from the continuity of the multiplication and the equation x · x−1 = e

that there exists a neighborhood Uδ(1)(x) with Uδ(1)(x)x−1 ⊆ U1(e). This fact

can be expressed as

∀z ∈ Uδ(1)(x) : ∃(zx−1)−1, i.e. zx−1(zx−1)−1 = e.

It is easy to see that x−1(zx−1)−1 is the inverse element of z. Therefore, the

statement above is proved.

Step 2: Continuity of the inverse function, i.e. x−1n → x−1

The series (zn)n∈N, defined by zn = xnx−1 converges to e, whenever xn con-

verges to x. Therefore, z−1n exists for all n with ‖zn − e‖ < 1. Furthermore,

z−1n = e+ (e− zn) + (e− zn)

2 + · · ·

and converges to e, whenever xn converges to x.

It is straightforward that the application (·)−1 is continuous on O.

x−1n = x−1xx−1

n = x−1z−1n

n→∞−−−−→ x−1e = x−1

17

Definition 2.6 ((Proper) ideal). A subset I of the algebra R is called ideal if

the following conditions are satisfied.

• If x and y are contained within I, then their sum x + y is an element of

I as well, i.e. x, y ∈ I ⇒ x+ y ∈ I.

• If x is an element of I and z an element of the algebra R, then x · z is an

element of I as well, i.e. x ∈ I ⇒ zx ∈ I ∀z ∈ R.

If in addition I 6= R, the ideal I is called a proper ideal of R.

Theorem 2.7 (Relation between inverse elements and proper ideals). An ele-

ment x of R has an inverse element x−1 ∈ R if and only if there does not exist

a proper ideal of R which contains x, i.e.

∃ x−1 ⇐⇒ x 6∈ Ip, ∀ proper ideals Ip of R.

Proof. “⇒”: If the inverse x−1 exists, x can not be contained within a proper

ideal because (z · x−1)︸ ︷︷ ︸∈R

·x ∈ I. Therefore, I coincides with R.

“⇐”: If x−1 does not exist, I = z · x|z ∈ R is a proper ideal because the

neutral element e is not contained within I.

Corollary 2.8 (Closure of a proper ideal). The closure I of a proper ideal I is

a proper ideal as well.

Proof. Due to theorem 2.5 and theorem 2.7, every proper ideal is contained

within R\O, which is as complement of on open set closed. Therefore, I belongs

to R\O as well.

Definition 2.9 (Maximal ideal). A maximal ideal M is a proper ideal which is

not contained within any other proper ideal of R.

Theorem 2.10 (Properties of maximal ideals).

• Every proper ideal I of R is contained within a maximal ideal of R.

18

• Every maximal ideal M of R is closed.

Proof. Step 1: The first part is proved by transfinite induction and is omitted,

see [13] theorem 11.3.

Step 2: As proper ideal the closure M of the maximal ideal M is also a proper

ideal. Since M does not belong to any larger proper ideal it must coincide with

M .

The following theorem concludes about the structure of a normed commutative

ring in terms of its maximal ideals.

Theorem 2.11 (Relation between inverse elements and maximal ideals). An

element x of R has an inverse element x−1 ∈ R if and only if there does not

exist a maximal ideal of R which contains x, i.e.

∃ x−1 ⇐⇒ x 6∈M, ∀ maximal ideals M of R

If the only maximal ideal of R is the trivial one, it follows that R is a field.

Proof. The statement follows directly from theorem 2.7 and theorem 2.10.

2.2.1 An example

Let Mτ be the set of all functions of V (l) which vanish in the point τ . Then,

Mτ is a maximal ideal in V (l). The same statement holds for any algebra of

functions.

The domain S of the functions in V (l) and its maximal idealsMτ are respectively

defined by

S := λ ∈ R | − π < λ ≤ π and

Mτ = z(λ) ∈ V (l) | ∃τ ∈ S : z(τ) = 0 .

Mτ is a maximal ideal:

It is obvious that Mτ is a proper ideal in R. Furthermore, every function z(λ)

in V (l) can be represented as

z(λ) =z(τ)

y(τ)y(λ) +

(z(λ)−

y(λ)

y(τ)z(τ)

).

19

The second summand is a member of Mτ . The first summand is a multiple of

y(λ) 6∈ Mτ . Hence, a proper ideal which contains Mτ and y(λ) 6∈ Mτ does not

exist.

2.3 Equivalence classes with respect to ideals

Definition 2.12 (Congruence with respect to an ideal I). Two elements x and

y of a commutative normed algebra R are congruent with respect to an ideal I

if and only if their difference is in I, i.e.

x, y ∈ R : x ∼ y ⇐⇒ x− y ∈ I.

Since this relation is reflexive, symmetric and transitive, R is partitioned in

classes of congruent elements. The quotient algebra of R with respect to I

is denoted by R/I. Consider for example the quotient algebra of continuous

functions on a compact set with respect to a maximal ideal, e.g. the maximal

ideal containing all the functions vanishing in one point. The equivalence classes

consist of those function which are equal in τ .

In the following, the equivalence class pertaining to x is denoted by [x]∼. By

contrast, small letters are used for the elements of the algebra.

Theorem 2.13 (R/I is a normed commutative algebra). Let R be a commuta-

tive normed algebra. The norm on R/I is defined by

‖[x]∼‖R/I = infx∈[x]∼

‖x‖R.

If I is a closed proper ideal, then R/I is a normed commutative algebra as well.

Proof. It is known from the Banach space theory that infx∈[x]∼

‖x‖ is a norm on

the quotient space of the Banach space R, see [13]. The multiplicativity of the

norm is evident as well.

Only the additional condition about the norm of the neutral element [e]∼ in

Banach algebras will be proved.

‖E‖R/I ≤ 1 : Since e is in [e]∼, the norm of [e]∼ has to be at most 1.

20

‖E‖R/I ≥ 1 : If y is contained within [e]∼, an element x ∈ I exists which sat-

isfies y = e + x. This is because x = y − e and because y and e are

in the same equivalence class [e]∼. If ‖y‖R were strictly smaller than 1,

the inverse element (y − e)−1 = x−1 of (y − e) would exist. This is a

contradiction to the fact that x is a member of the proper ideal I.

Theorem 2.14. The homomorphism HI

HI :

R → R/I

x 7→ [x]∼

assigns to every element x ∈ R the equivalence class [x]∼ with respect to I. For

every closed ideal I, HI is an open and continuous application.

Proof. Step 1: HI is open.

Let Uδ(0) = x ∈ R | ‖x‖R < δ ⊆ R be an open ball cent red at zero. Due to

the definition of the norm on R/I, the image U ′ = HI(x) ∈ R/I : x ∈ Uδ(0)

of Uδ(0) in R/I consists of all equivalence classes with ‖[x]∼‖ < δ. Thus, the

image U ′ is an open subset in R/I.

This means in mathematical terms that

‖HI(x)‖R/I = ‖[x]∼‖R/I = infx∈[x]∼

‖x‖R ≤ ‖x‖R < δ.

Since the open balls constitute a basis of the topology of R, the image of every

open subset in R is an open subset in R/I.

Step 2: HI is continuous, i.e. if F ′ is closed in R/I, then the pre-

image F = H−1I (F ′) is closed in R as well.

Let (xn)n∈N be a Cauchy sequence in F with limit x in R. Furthermore, xi and

the limit point x are respectively contained within the corresponding equiva-

lences [xi]∼ and [x]∼. It must be shown that limn→∞

xn = x is contained within

F .

From the inequality ‖[x]∼ − [xn]∼‖ = inf(x−xn)∈[x−xn]∼

‖x − xn‖ ≤ ‖x − xn‖, it

follows that the corresponding equivalence classes [xn]∼ satisfy

limn→∞

[xn]∼ = [x]∼.

21

Furthermore, [x]∼ is contained within F ′ ⊆ R/I, because F ′ is closed. Now, the

arbitrarily chosen Cauchy sequence (xn)n∈N ∈ FN converges towards an element

x ∈ F . Therefore, F is a closed subset in R.

Theorem 2.15. For J ⊇ I, there exists a bijective relation between the closed

ideals J of R and J ′ of R/I.

Proof. Step 1: If J’ closed, then J closed and H−1I is injective.

Since HI is continuous, the pre-image J ⊆ R of a closed ideal J ′ ⊆ R/I is a

closed ideal in R. Obviously, J contains I.

Since

J = H−1I (J ′) = j + i ∈ R : HI(j) = [j]∼ ∈ J

′ ∧ i ∈ I,

it follows that for a [j2]∼ contained within J ′2 but not within J ′1 the correspond-

ing element j2 ∈ R lies in J2 but not in J1. Therefore, it results from the

inequality J ′1 6= J ′2 that J1 = H−1I (J ′1) 6= J2 = H−1

I (J ′2) as well. Thus, H−1I is

injective.

Step 2: If J is closed, then J ′ is closed as well.

The image J ′ of an ideal J containing I is an ideal in the algebra R/I. Further-

more, J is the pre-image of J ′ because with an element x the whole equivalence

class [x]∼ is contained within J . If the pre-image of an open map is closed, the

image is closed as well. Since the homomorphism HI is open, it follows that J ′

is closed in R/I.

The proper ideals of R/I are the images of the proper ideals of R. In particular,

the maximal ideals of R/I are the images of the maximal ideals of R.

Theorem 2.16. The quotient algebra R/M of a commutative normed algebra

R with respect to a maximal Ideal M is a field.

Proof. In order to prove this, it will be shown that there is no proper ideal in

R/M which is not trivial (see theorem 2.11 and theorem 2.7). If J were a non-

trivial ideal in R/M , its pre-image would be a proper ideal in R which contains

M but does not coincide with M . This is a contradiction to the maximality of

M .

22

The following theorem establishes that the other direction is true as well.

Theorem 2.17. If the quotient algebra R/I of a commutative algebra R with

respect to a proper ideal I is a field, then I is a maximal ideal of R. (The

assumption that I is closed is not necessary.)

Proof. Assuming that J is a proper ideal of R which contains I and does not

coincide with I it follows that its image HI(J) is a non-trivial proper ideal

of R/I. Therefore, there exists an element [x]∼ in the proper ideal HI(J)

which isn’t the zero element. Due to theorem 2.7, this is equivalent to the fact

that the inverse element of [x]∼ does not exist. This is a contradiction to the

assumption.

2.4 The canonical isomorphism on the maximal ideals

Definition 2.18 (Spectrum of x). Let x be a member of a normed algebra R.

The set

σ(x) = λ ∈ C| 6 ∃(x − λe)−1

is called the spectrum of the element x.

Theorem 2.19. Every element x of R has a non-trivial spectrum.

Proof. Assume that the element x ∈ R has an empty spectrum, i.e. (x− λe)−1

exists for all complex numbers λ. For λ = 0 the existence of x−1 follows.

The element x−1 has an empty spectrum as well, i.e. ∃(x−1 − λe)−1 ∀ λ ∈ C.

For λ = 0, this is obvious. For λ 6= 0 the existence follows from the expression

(x−1 − λe)−1 =

[(1

λe− x)(x−1λ)

]−1

= −1

λx(x −

1

λe)−1.

Therefore, (x−1 − λe)−1 and (x− λe)−1 are entire vector functions in λ. Their

Taylor series∞∑n=0

xn+1λn and∞∑n=0

x−n−1λn are absolutely convergent in the com-

plex plane, in particular for λ = 1. It follows that ‖xn‖ and ‖x−n‖ converge

towards 0. This is a contradiction to

1 = ‖e‖ = ‖xnx−n‖ ≤ ‖xn‖‖x−n‖.

23

Theorem 2.20. A normed field (a normed commutative division algebra) R is

isomorphic to the field of complex numbers.

Proof. Due to the last theorem, there exists a λ ∈ C for every x ∈ R such that

(x− λe) is not invertible in R. Since R is a field, it follows that x−λe = 0.

The application

Iso :

R → C

λe 7→ λ

defines the isomorphism between the normed field and the complex plane.

Since the quotient algebra with respect to a maximal ideal is a field, the Gelfand-

Mazur theorem follows.

Theorem 2.21. The quotient algebra of a commutative normed algebra R with

respect to a maximal ideal M is isomorphic to the field of complex numbers.

2.4.1 The relation between maximal ideals and complex multiplica-

tive homomorphisms

Theorem 2.22.

• To every maximal ideal M corresponds a complex multiplicative homomor-

phism from the algebra R in the field of complex numbers.

• To every non-trivial homomorphism from the algebra R in the field of

complex numbers corresponds a maximal ideal.

Proof. First part.

The aforementioned homomorphism is defined by

H = (Iso HM ) :

RHM−−→ R/M

Iso−−→ C

x 7→ [x]∼ 7→ λ.

Second part.

The kernel of a given homomorphismH is an ideal in R. Due to the fundamental

theorem on homomorphisms, the factor algebra R/ ker(H) is isomorphic to the

image of the homomorphism. Since the image of the non-trivial homomorphism

24

H is a non-empty subring of the complex number field, the image H(R) must

coincide with C. If H : R → C is a non-trivial homomorphism, it follows by

theorem 2.17 that

• R/M is a field and that

• there exists a maximal ideal M satisfying ker(H) = M .

Definition 2.23 (The function x(M) on the maximal ideal space). Let x be an

element of the normed commutative algebra R. The function

x :

M(R) → C

M 7→ (Iso HM )(x)

assigns to every maximal ideal M in the space of maximal ideals M(R) of R

the complex number (Iso HM )(x).

For fixed x and variable maximal ideal M , x(M) is a function on the set of all

maximal ideals M =M(R) of R. The set of all these applications is denoted

with R.

Properties of the function x(M)

(i). (x1 + x2)(M) = x1(M) + x2(M) ∀M ∈M

(ii). (x1 · x2)(M) = x1(M) · x2(M) ∀M ∈M

(iii). (λx1)(M) = λ(x1)(M) ∀M ∈M

(iv). e(M) ≡ 1 ∀M ∈ M

(v). x is contained within M if and only if x(M) = 0.

(vi). If M1 6= M2, there exists an x ∈ R which separates M1 and M2, i.e.x(M1) 6= x(M2).

• This separability condition shows that different maximal ideals pro-vide different linear functionals.

(vii). |x(M)| ≤ ‖x‖ ∀M ∈ M

25

• x(M) is that number λ[x]∼ which the canonical isomorphism betweenR/M and the complex number field assigns to the equivalence class[x]∼ containing x. Since [x]∼ = λ[x]∼ [e]∼, it follows that

‖[x]∼‖R/I = |λ[x]∼ |‖[e]∼‖R/I = |λ[x]∼ |

Considering the definition of the norm leads to

x(M) = |λ[x]∼ | = ‖[x]∼‖R/I = infz∈[x]∼

‖z‖ ≤ ‖x‖.

The functions x(M) form an algebra R with neutral element.

The assertion follows by properties 1 - 4. The application

(·) :

R → R

x 7→ (M 7→ x(M))

is an homomorphism from R to R.

M(x) defines a linear bounded multiplicative functional with norm 1.

By properties 1,2,3,4 and 7, M(x) defines for fixed M ∈ M and variable x ∈ R a

linear bounded multiplicative functional on R. These linear functionals are to be

distinguished from normal linear functionals by their multiplicativity property

M(x1x2) = M(x1)M(x2),

which follows from 2.

Due to property 5, the condition for the existence of an inverse element and

hence theorem 2.11 can be reformulated.

Theorem 2.24. An element x of R has an inverse if and only if x(M) does

not vanish on M(R).

Theorem 2.25. The spectrum of x coincides with the image of x(M), i.e.

σ(x) = Im(x(M)).

Proof. ⊇: If x(M0) = λ0, then (x − λ0e)(M0) = 0. By property 5, this is

equivalent to (x−λ0e) ∈M0. Therefore, (x−λ0e) has no inverse element.

⊆: If (x−λ0e)−1 does not exist, (x−λ0e)(M) vanishes for a maximal ideal M0,

i.e. x(M0) = λ0.

26

2.5 Proof of the univariate case

Theorem 2.26. Let f(ζ) be an analytic function on a superset of the spectrum

of an element x ∈ R. Let Γ be a rectifiable, simple-closed Jordan-curve, which

lies in the area of regularity of f(ζ) and encloses σ(x).

Under this assumptions the following points hold:

• The integral1

2πi

∫

Γ

f(λ)

(λe− x)dλ = f(x) exists and is independent of the

choice of Γ.

• The canonical homomorphism

(·) :

R → R

x 7→ (M 7→ x(M))

maps the element f(x) of R to the element f(x(M)) of R, which is defined

by ˜f(x)(M) = f(x(M)).

• With x(M), f(x(M)) is a member of R as well.

Proof. Since the spectrum of x is enclosed by Γ, the function z(λ) = (λe −

x)−1f(λ) is well defined at any point of Γ. Furthermore, z(λ) is continuous

with respect to the chosen norm. The integral exists in the sense of norm-

convergence and it is independent of the choice of Γ. This is why Cauchy’s

integral theorem can be applied. For fixed M and variable x, x(M) is a linear

functional in x. Therefore,

˜f(x)(M) =1

2πi

∫

Γ

(λe− x)−1(M)f(λ)dλ

=1

2πi

∫

Γ

f(λ)

(λ − x(M))dλ = f(x(M)).

2.6 The topology on M(R)

In order to prove the multivariate case, some topological properties of the space

M(R) are required.

In this section, it will be shown that there exists a compact (and unique) Haus-

dorff topology on the space of maximal idealsM(R) of an algebra R.

27

For arbitrarily chosen elements x1, x2, . . . , xn of the algebra R and for every

ε > 0, a neighborhood of the point M0 ∈ M(R) is defined by

Ux1,...,xnε (M0) = M ∈ M(R) | |x1(M)− x1(M0)| < ǫ, . . . , |xn(M)− xn(M0)| < ǫ .

These neighborhoods define a fundamental system of neighborhoods which

uniquely determines the topology ofM(R).

Definition 2.27 (Fundamental system of neighborhoods B(x)). Let X be a

non-empty set and x an arbitrarily chosen point in X. A non-empty subset

B(x) of P(X) is called fundamental system of neighborhoods of x, if the following

conditions hold.

(i). U1, U2 ∈ B(x)⇒ ∃V ∈ B(x) : V ⊆ U1 ∩ U2

(ii). U ∈ B(x)⇒ x ∈ U

(iii). U ∈ B(x)⇒ ∃V ∈ B(x) : ∀y ∈ V ∃W ∈ B(y) : W ⊆ U

The corresponding topology τ of X is defined as

τ := G ∈ P(X) | ∀x ∈ G ∃U ∈ B(x) : U ⊆ G

Evidently, B(M0) = Ux1,...,xnε (M0) | xi ∈ R, ∀i ∈ 1, . . . , n, ε > 0 is non-

empty and every Ux1,...,xnε (M0) contains M0.

Because of the following conclusion, conditions (1) and (3) are satisfied as well.

• Ux1,...,xn,xn+1,...,xn+m

minε1,ε2(M0) ⊆ Ux1,...,xn

ε1(M0) ∩ Uxn+1,...,xn+m

ε2 (M0)

• If M1 is an element of Ux1,...,xnε (M0), it follows that Ux1,...,xn

δ (M1) ⊆

Ux1,...,xnε (M0), where 0 < δ < minε−|x1(M1)−x1(M0)|, . . . , ε−|xn(M1)−

xn(M0)|.

Since x separates points inM(R), i.e.

M ′ 6= M ⇒ ∃x ∈ R : x(M ′) 6= x(M),

the topology ofM(R) is Hausdorff, i.e.

Uxε2(M) ∩ Uxε

2(M ′) = ∅ ∀M ′ : ε < |x(M ′)− x(M)|

28

By the definition of the topology onM(R), x(M) is continuous, i.e.

Uxε (M0) = M ∈ M(R) | |x(M)− x(M0)| < ε

⇐⇒

M ∈ Uxε (M0)⇒ x(M) ∈ UCε (x(M0)),

where UCε (x(M0)) denotes an open ball with cent re x(M0) and radius ε in C.

In order to prove that the topology on M(R) is compact, some topological

fundamentals are required. They can be found in [13].

Lemma 2.28. Let (X, τ) be a topological Hausdorff space and A ⊆ X a compact

set. Then, it follows that A is closed.

Lemma 2.29. If f : (X1, τ1)→ (X2, τ2) is continuous and A ⊆ X1 compact, it

follows that f(A) ⊆ X2 is compact as well.

Lemma 2.30. If f : (X1, τ1) → (X2, τ2) is continuous and bijective, where

(X1, τ1) is compact and (X2, τ2) Hausdorff, it follows that f−1 is continuous.

Lemma 2.31 (Tychonoff). If (Xi, τi), i ∈ I is a family of compact topological

spaces, it follows that (∏i∈I

Xi,∏i∈I

τi) is a compact topological space as well.

The main statement of this subsection can now be proved.

Theorem 2.32. The topological Hausdorff space (M(R),) is compact.

Proof. Step 1: Preparations

A compact disc Qx ⊆ C is assigned to every x ∈ R.

• x 7→ UC

‖x‖(0) =: Qx

By Tychonoff’s theorem, the Cartesian product Q of all Qx,

Q :=∏

x∈R

Qx,

is, endowed with the product topology τ , a compact space. The corresponding

fundamental system of neighborhoods of a point (λox)x∈R ∈ Q is given by

(λox)x∈R ∈ Q | |λx1 − λ

ox1| < ε, . . . , |λxn − λ

oxn | < ε

, ε > 0;x1, . . . , xn ∈ R

Step 2: The mapping f :

M(R) → Q|f(M(R))

M 7→ (µx)x∈R = (x(M))x∈Ris well-

defined and bijective.

29

Well-defined: Since |x(M)| ≤ ‖x‖, it follows that

∀M ∈M(R) : ∃(µx)x∈R ∈ Q,µx = x(M) ∀x ∈ R

Injectivity: For every pair of non-equal maximal ideals exists a x′ ∈ R, which

assigns different values to those different maximal ideals. Therefore, the

corresponding points in Q do not coincide either, at least in the coordinate

x′, i.e.

M1 6= M2 ⇒ ∃x′(M1) 6= x′(M2)

⇒ (µx)1x∈R 6= (µx)

2x∈R, because µ1

x′ = x′(M1) 6= x′(M2) = µ2x′ .

Step 3: The application f is a homeomorphism

A comparison between the fundamental systems of neighborhoods ofM(R) and

Q shows that f and f−1 are continuous.

M(R) : Ux1,...,xnε (M0) = M ∈ M(R) | |x1(M)− x1(M0)| < ǫ, . . . , |xn(M)− xn(M0)| < ǫ

Q : Ux1,...,xnε ((λox)x∈R) = (λox)x∈R ∈ Q | |λx1 − λ

ox1| < ǫ, . . . , |λxn − λ

oxn | < ǫ

Step 4: M′ = f(M(R)) is closed in the topology of Q

It has to be shown that M′ is closed, because then (M(R),) is compact

and the theorem is proved. If M′ is closed, then it is as closed subset of a

compact Hausdorff space compact too. Since f is a homeomorphism,M(R) =

f−1(f(M(R))) is compact as well.

In order to show thatM′ is closed, it will be proved that if Λ = (λox)x∈R is an

element of the closureM′ ofM′, Λ belongs toM′ as well, i.e. the equations

λox+y = λox + λoy, cλox = λocx and λoxy = λoxλ

oy.

hold. Since Λ = (λox)x∈R is contained withinM′, there exists a neighborhood

V :=

(λx)x∈R ∈ R | |λe − λoe| < ǫ, |λx − λ

ox| < ǫ, |λy − λ

oy| < ǫ, |λxy − λ

oxy| < ǫ

=

= Ue,x,y,xyε (Λ)

of Λ = (λox)x∈R whose intersection with M′ is non-empty and contains an

element (λ′x)x∈R. Furthermore, there exists a maximal ideal M whose picture

is (λ′x)x∈R because the function f is bijective.

30

Λ ∈M′ ⇒ ∃(λ′x)x∈R ∈ V ∩M′ ∧ M ∈M(R) : f(M) = (λ′x)x∈R

|λoe − e(M)︸ ︷︷ ︸=λ′e

| = |λoe − 1| < ε

|λox − x(M)︸ ︷︷ ︸=λ′x

| < ε, |λoy − y(M)︸ ︷︷ ︸=λy

| < ε

|λoxy − xy(M)︸ ︷︷ ︸=λ′xy

| = |λoxy − x(M)y(M)| < ε

Only the multiplication property, λoxy = λoxλoy , will be proved.

|λoxy − λoxλoy | ≤ |λ

oxy − x(M)y(M) + x(M)y(M)− x(M)λoy + λoyx(M) − λoyλ

ox|

≤ |λoxy − x(M)y(M)|+ ‖x‖|y(M)− λoy|+ |λoy||x(M)− λox|

< ε(1 + ‖x‖+ |λoy |)

The other mentioned properties are proved in the same manner.

Therefore, there exists a homomorphism

H :

R → C

x 7→ λox

with ker(H) = M ′, where x(M ′) = λx for every x ∈ R.

2.7 Generating systems and polynomially convex sets

Definition 2.33 (Generating system). A subset K ⊆ R is a generating system

of the algebra R if and only if the smallest closed algebra with neutral element

of multiplication which contains K is R.

• The neutral element is not a generating element.

Theorem 2.34 (Finite number of generating elements). If the algebra R has

a finite number of generating elements, the space M(R) is homeomorph to a

compact set F ⊆ Cn.

Proof. The set F = z ∈ Cn | ∃M ∈M(R), z = (x1(M), . . . , xn(M)) is com-

pact because it is the image of a compact set.

31

The application

x1

...

xn

: (M(R),) −→ (F, ‖·‖2)

is a continuous and bijective mapping from a compact space into a Hausdorff

space and therefore a homeomorphism.

The injectivity is an implication of the following reasoning. If xi(M1) = xi(M2),

i ∈ 1, . . . , n, all the polynomials in x1, . . . , xn coincide as well, i.e.

xi(M1) = xi(M2), ∀i⇒

P (x1(M1), . . . , xn(M1)) = P (x2(M1), . . . , xn(M2)),

for all polynomials in x1, . . . , xn.

Since x1, . . . , xn is a generating system of R, it follows that

x(M1) = x(M2) ∀x ∈ R.

By the separation property, M1 must coincide with M2.

Definition 2.35 (Polynomially convex set). A subset F of Cn is called poly-

nomially convex if and only if for every ζ′ in F , there exists a polynomial

P (ζ1, . . . , ζn) which satisfies the conditions

• P (ζ′) = 1

• |P (ζ)| < 1, ∀ζ ∈ F .

Theorem 2.36. F ⊆ Cn is homeomorph to the space of maximal ideals M(R)

of an algebra R with n generating elements if and only if F is closed, bounded

and polynomially convex.

Proof. “⇒”: Let R be an algebra generated by n elements zj , j ∈ 1, . . . , n.

Then F = ζ ∈ Cn | ∃M ∈ M(R) : ζ = (x1(M), . . . , xn(M)) is closed and

bounded becauseM(R) is compact.

Polynomial convexity:

ζ′ 6∈ F ⇐⇒ 6 ∃H = (Iso HM ) : H(xj) = ζ′j ∀j ∈ 1, . . . , n

⇐⇒ 6 ∃M ∈ M(R) : (xj − ζ′je)(M) = 0 ∀j ∈ 1, . . . , n

⇐⇒ 6 ∃M ∈ M(R) : xj − ζje ∈M ∀j ∈ 1, . . . , n

32

Therefore, the idealn∑

j=1

(xj − ζ′je)rj , rj ∈ R

isn’t contained within any maximal ideal M ∈ M(R) and coincides with the

whole algebra.

Since the neutral element of multiplication is a member of every algebra R, it

follows that there exist qj , j ∈ 1, . . . , n, such that

e =n∑

j=1

(xj − ζ′je)qj.

These qj can be approximated, with arbitrary accuracy, by polynomials in xi, i ∈

1, . . . , n, i.e.

∀ǫ > 0 ∃Pj : ‖e−n∑

j=1

(xj − ζ′je)Pj(x1, . . . , xn)‖ < ε.

The corresponding polynomial in (ζ1, . . . , ζn), defining the polynomially convex

set, is

1−n∑

j=1

(ζj − ζ′je)Pj(ζ1, . . . , ζn).

“⇐”: It will be shown that a closed, bounded and polynomially convex subset

F of Cn is homeomorph to a space of maximal ideals of an algebra R.

Let R′ be the algebra of all polynomials P (ζ1, . . . , ζn), which is endowed with

the norm ‖P‖ = maxζ∈F|P (ζ1, . . . , ζn)|. R is the completed version of R′ (with

respect to ‖·‖) and the zj(ζ) = ζj , j ∈ 1, . . . , n, are the generating elements

of R.

Since the algebra R has n generating elements, a compact set F ′, which is home-

omorph to M(R), exists. Due to the following reasoning, the set F coincides

with F ′.

F ⊆ F ′ : Every ζ0 ∈ F defines by the homomorphism zj(ζ) 7→ ζ0j a maximal

ideal in R. Therefore, ζ0 is contained within F ′ ∼=M(R) as well.

Injectivity: Different points in F give different maximal ideals.

ζ1 6= ζ2 ⇒ ∃j ∈ 1, . . . , n : H1j (zj) = ζ1

j 6= ζ2j = H2

j (zj)⇒Mζ1 6= Mζ2

33

F ⊇ F ′ : If ζ′ is not contained within F , there exists a polynomial whose ab-

solute value is smaller than one for every ζ in F and whose value at ζ′

is one. Since F is compact, the norm of the polynomial is smaller than

one. Therefore, a maximal ideal which corresponds to ζ′ does not exist.

In mathematical terms, this means

ζ′ 6∈ F ⇒ ∃P (ζ1, . . . , ζn) : P (ζ) < 1 ∀ζ ∈ F, ∧ P (ζ′) = 1

⇒ ‖P‖ = maxζ∈F|P (ζ)| = c < 1

⇒ |P (z1, . . . , zn)(M)| ≤ ‖P (z1, . . . , zn)‖ = c < 1⇒ ζ′ 6∈ M(R) ∼= F ′.

34

2.8 Proof of the multivariate case

The purpose of this section is the proof of the following theorem.

Theorem 2.37. If f(ζ1, . . . , ζn) is analytic on the set

σR(x1, . . . , xn) = ζ ∈ Cn | ∃M ∈ M(R) : ζ = (x1(M), . . . , xn(M)) ,

then there exists an x ∈ R satisfying

f(x1(M), . . . , xn(M)) = x(M), ∀M ∈M(R).

In a first step, it will be proved for the case where the algebra is generated

by a finite number of elements. Subsequently, this result is generalized to an

arbitrary normed commutative algebra.

2.8.1 Case A: Algebra R is generated by a finite number of elements

In analogy to the proof of the one dimensional case, the analytically transformed

element will be represented as a multidimensional integral on a certain domain.

The domain contains the compact and polynomially convex set, which is iso-

morphic to the space of maximal ideals. Since the algebra is complete, this

integral is a member of the algebra as well. The integral representation will be

proved by Weil’s theorem 2.39.

Definition 2.38 (Weil domain). The finite intersection of sets of the form Gν =

ζ ∈ Cn | |Pν(ζ1, . . . , ζn)| < 1, i.e. the set G =N⋂ν=1

Gν , is a Weil domain if and

only if the real dimension ofn⋂k=1

ζ ∈ Cn | |Pik(ζ1, . . . , ζn)| = 1 is smaller than

n for all i1, . . . , in ⊆ 1, . . . , N.

Weil’s theorem

The Weil domain is defined by inequalities of the form

|Pi(ζ1, . . . , ζn)| < 1, i ∈ 1, . . . , N.

The intersection of the boundary of n such domains,n⋂j=1

ζ ∈ Cn | |Pij (ζ)| = 1

,

is denoted by σi1···in . This intersection is endowed with a certain orientation.

35

For every point (τ1, . . . , τn) ∈ F , the difference of two polynomials can be rep-

resented as

Pi(ζ1, . . . , ζn)− Pi(τ1, . . . , τn) =n∑

j=1

(ζj − τj)Qij(ζ1, . . . , ζn),

where the Qij are polynomials in (ζ1, . . . , ζn) and its coefficients depend on

(τ1, . . . , τn).

Theorem 2.39 (Weil’s integral representation). Let f(ζ1, . . . , ζn) be an analytic

function on a domain G ⊆ Cn which contains a Weil domain G. Furthermore,

τ = (τ1, . . . , τn) is an element of F ⊆ G and Di1···in denotes det[(Qiαj)

nα,j=1

].

Then, this function can be represented as

f(τ1, . . . , τn) =

N∑

i1,...,in=1i1<···<in

(1

2πi

)n ∫

σi1···in

Di1···inf(ζ1, . . . , ζn)n∏j=1

Pij (ζ1, . . . , ζn)− Pij (τ1, . . . , τn)dζ1 · · · dζn.

Proof. See B.A. Fuks [5].

Interpretation in the algebra

• Since Pij (ζ1, . . . , ζn)e − Pij (x1, . . . , xn) does not vanish for

ζ ∈ σi1···in and τ is contained within F , it follows that[Pij (ζ1, . . . , ζn)e− Pij (x1, . . . , xn)

]−1exists as continuous function

in ζ.

• Since Di1···in is a polynomial in ζ for fixed τ or fixed elements x1, . . . , xn ∈

R, Di1···in(ζ)f(ζ)

[n∏j=1

Pij (ζ1, . . . , ζn)e− Pij (x1, . . . , xn)

]−1

can be inte-

grated over σi1···in with respect to ζ.

The integral is a member of R as well, because the algebra is complete.

Since the canonical homomorphism HM is continuous, the values of the ob-

tained element in R coincide on M ∼= F with the values of f(τ1, . . . , τn) =

f(x1(M), . . . , f(xn(M))) in the corresponding points of F ∼=M(R).

In order to apply Weil’s theorem, the Weil domain will now be constructed.

36

Construction of the Weil domain.

M(R) is homeomorph to a compact and polynomially convex subset F of Cn.

The function f(ζ) is analytic on an open superset G of F which can be written

as intersection of sets of the form

Gν = ζ ∈ Cn | |Pν(ζ1, . . . , ζn)| < 1 ,

because it is polynomially convex. The Weil domain is the intersection of a

finite number of sets from F =∞⋂ν=1

Gν . These sets will be chosen such that the

Weil domain is contained within the area of regularity of f(ζ) and that the real

dimension of its boundary is smaller than n for all possible choices of n indexes.

The Weil domain G will be constructed by a compactness argument. Since F

is compact, the maxima mν of the polynomials Pν exist and they are smaller

than 1. Hence, F can be represented as intersection of closed, even compact sets

Fν by choosing a constant ϑν bigger than the maximum mν of the polynomial

Pν(ζ) and smaller than 1.

• M(R) ∼= F ⊆ Fν ⊆ Gν

maxζ∈F|Pν(ζ1, . . . , ζn)| = mν < ϑν < 1

Fν = ζ ∈ Cn | |Pν(ζ1, . . . , ζn)| ≤ ϑν

Gν = ζ ∈ Cn | |Pν(ζ1, . . . , ζn)| < 1

∗ Since F =⋂νGν and Fν ⊆ Gν it follows that F =

⋂νFν .

In the same manner, the generating elements assume their maxima. Therefore,

the sets Qj and Sj , created from the generating elements, can be defined.

• M(R) ∼= F ⊆ Sj ⊆ Qj

maxM∈M(R)

|zj(M)| = maxζ∈F|ζj | < aj < cj

Sj = ζ ∈ Cn | |ζj | ≤ aj

Qj = ζ ∈ Cn | |ζj | < cj

Since the complement Gc of G and the sets Fν are closed in Cn, they are closed

in the relative topology of the compact set S =n⋂j=1

Sj as well. If the intersection

37

of a family of sets in a compact space is empty, a finite number of sets can be

chosen, whose intersection is empty too, i.e.

⋂

ν

Fν ∩Gc ∩ S = ∅ =⇒ ∃ν1, . . . , νn :

m⋂

i=1

Fνi ∩Gc ∩ S = ∅

=⇒m⋂

i=1

Fνi ∩ S ⊆ G.

Therefore, the Weil domain G in demand is

G =m⋂

i=1

F oνi ∩n⋂

j=1

Soj ,

where Ao denotes the interior of A. In order to prove that G is a Weil domain,

the second condition in its definition must be verified. It must be shown that

the real dimension of the boundary is smaller than n.

The real dimension of the boundary is smaller than n, i.e. the Pν

can be chosen not being functional dependent:

The polynomial Pν(ζ) is defined by

Pν(ζ) =1

ϑνPν(ζ) +Rν(ζ),

where the absolute value of Rν(ζ) is not bigger than ε on

Q =n⋂

j=1

Qj = ζ ∈ Cn | |ζj | < cj , j ∈ 1, . . . , n ⊇ F.

It will be shown that an ε > 0 and polynomials Pν(ζ) exists, such that the inter-

section of Q and a set generated by Pν(ζ) is contained within the set generated

by Pν(ζ), i.e.

F ⊆(ζ ∈ C

n | |Pν(ζ)| < 1∩Q)⊆ ζ ∈ C

n | |Pν(ζ)| < 1 , ∀ ν.

First Inclusion: Evidently, an ε > 0 can be chosen such that the norm of

Pν(ζ) is smaller than one.

maxζ∈F|Pν(ζ)| ≤

1

ϑνmaxζ∈F|Pν(ζ)|+ max

ζ∈F|Rν(ζ)| ≤

mνϑν︸︷︷︸<1

+ε

Hence, if ζ is an element of F , it is an element ofζ ∈ Cn | |Pν(ζ)| < 1

∩

Q as well.

38

Second Inclusion: If the norm of Pν(ζ) is smaller than 1, it follows that an

ε > 0 exists such that Pν(ζ) is smaller than one as well.

‖Pν(ζ)‖ = ‖ϑν(Pν(ζ) −Rν(ζ))‖ < ϑν(1 + ε)

Hence, if ζ is an element ofζ ∈ Cn | |Pν(ζ)| < 1

∩ Q, it is contained

within ζ ∈ Cn | |Pν(ζ)| < 1 as well.

Therefore, one can chose the polynomials Pν(ζ) in a way that they are not

functionally dependent on each other and that the real dimension of

n⋂

k=1

ζ ∈ Cn | | ˜Pik(ζ1, . . . , ζn)| = 1

, i1, . . . , in ⊆ 1, . . . , N

is smaller than n.

2.8.2 General Case

Definition 2.40 (Common spectrum). The set

σR(x1, . . . , xn) = ζ ∈ Cn | ∃M ∈ M(R), ζ = (x1(M), . . . , xn(M))

is called the common spectrum of n elements x1, . . . , xn of R..

• σR(x1, . . . , xn) is the image of a compact set and therefore com-

pact as well. Moreover, the common spectrum is contained within

ζ ∈ Cn | |ζj | ≤ ‖xj‖, j ∈ 1, . . . , n.

• When R0 ⊆ R is the subalgebra generated from x1, . . . , xn, in general,

only the inclusion

σR(x1, . . . , xn) ⊆ σR0 (x1, . . . , xn)

holds. This is because the intersection of the maximal ideals of R with R0

is a maximal ideal in R0.

• If the elements x1, . . . , xn generate R, σR(x1, . . . , xn) coincides with

M(R) ∼= F . Thus, theorem 2.37 follows directly from Case A.

For every analytic function f(ζ) on σR(x1, . . . , xn) an open set U ⊇

σR(x1, . . . , xn) exists, where f(ζ) is analytic as well.

39

It will be proved that a subalgebra R′ ⊆ R exists which is generated by finitely

many elements x1, . . . , xn, y1, . . . , yp such that the projection σR′(x1, . . . , xn) ⊇

σR(x1, . . . , xn) of the common spectrum σR′(x1, . . . , xny1, . . . yp) ⊆ Cn+p on Cn

is contained within the open set U as well. Firstly, it will be shown that for every

point, which isn’t contained within σR(x1, . . . , xn), a whole neighborhood of this

point isn’t contained within σR1 (x1, . . . , xn), where R1 ⊆ R′ ⊆ R. Secondly, a

compactness argument will be used to construct R’ such that σR′(x1, . . . , xn) ⊆

U .

Then, f(ζ) can be continued constantly acrossM(R′) ⊇M(R) on the remaining

p arguments. Due to case A, there exists an x in R′ with the property in demand.

Step 1: Choice of y1, . . . , yp.

The circumstance that ζ′ is not an element of σR(x1, . . . , xn) is equivalent to the

fact that there is no maximal ideal in R, which contains all differences xj − ζ′je,

j ∈ 1, . . . , n, i.e.

ζ′ 6∈ σR(x1, . . . , xn) ⇐⇒ 6 ∃H = (Iso HM ) : H(xj) = ζ′j ∀j ∈ 1, . . . , n

⇐⇒ 6 ∃M ∈ M(R) : (xj − ζ′je)(M) = 0 ∀j ∈ 1, . . . , n

⇐⇒ 6 ∃M ∈ M(R) : xj − ζ′je ∈M ∀j ∈ 1, . . . , n.

Therefore, no maximal ideal contains the ideal

n∑

j=1

(xj − ζ′je)qj , qj ∈ R,

generated by

(xj − ζ′je) | j ∈ 1, . . . , n

. Hence, this ideal coincides with the

whole algebra.

Furthermore, there exist elements y1, . . . , yn ∈ R such that

e =n∑

j=1

(xj − ζ′je)yj.

In a neighborhood UCn

ε (ζ′) the difference between the neutral element and∑nj=1(xj − ζje)yj is smaller than 1 for every ζ ∈ UC

n

ε (ζ′). For this reason,

it has an inverse element, i.e.

∃UCn

ε (ζ′) : ‖e−n∑

j=1

(xj − ζje)yj‖ < 1, ∀ζ ∈ UCn

ε (ζ′).

40

Now, since e, x1, . . . , xn, y1, . . . , yp and their powers are already contained within

the subalgebra R1 ⊆ R, generated by these elements, the inverse element is

even contained within R1. By theorem 2.11, it follows that (x − ζe) isn’t a

member of any maximal ideal M ofM(R1). This is equivalent to the fact that

x(M) 6= ζ for all maximal ideals M inM(R1) and that ζ isn’t contained within

σR1 (x1, . . . , xn).

Therefore,

ζ 6∈ σR1 (x1, . . . , xn), ∀ζ ∈ UCn

ε (ζ′)

and hence

ζ 6∈ σR′ (x1, . . . , xn), ∀ζ ∈ UCn

ε (ζ′), ∀R′ ⊇ R1.

Step 2: Compactness argument.

The set theoretic difference Z\U between

Z = ζ ∈ Cn | |ζj | ≤ ‖xj‖, j ∈ 1, . . . , n

and U is compact and can be covered by finitely many of the above constructed

neighborhoods UCn

εi (ζ′(i)).

R′ is the algebra generated by the elements x1, . . . , xn, y1, . . . , yn, . . . , yp which

belong to the finite number of the open balls UCn

εi (ζ′(i)) in the finite cover

of Z\U . Due to the above reasoning, the projection of the common spec-

trum of x1, . . . , xn, y1, . . . , yn, . . . , yp on the space of the first n components, i.e.

σR′(x1, . . . , xn) does not contain any point of Z\U and is therefore contained

within U .

41

3 Perturbation of Polynomials

In this chapter, the behavior of polynomials under perturbation of their coeffi-

cient vector will be investigated. In particular, the question whether the roots

of the characteristic polynomial depend continuously on the coefficients will be

answered positively. The whole theory is developed in the frame of complex

analysis. Firstly, the structure of critical points will be investigated. Critical

points are those points for which the number of distinct roots decreases. Sec-

ondly, the notion of cycles of roots will be introduced. The cycles, which evolve

from analytic continuation of roots, will play a crucial role in the behavior of

the corresponding eigenprojections.

Furthermore, it has been shown in Kato [11], that the eigenvalues can be rep-

resented as continuous functions of a real parameter if the coefficients of the

(complex) matrix depend continuously on it. In the case of a complex pa-

rameter, the representation as continuous or analytic function of that complex

parameter is in general not possible if multiple roots occur.

Definition 3.1 (Polynomial of order n). The function

p(s, a) = ansn + an−1s

n−1 + · · ·+ a0 ∈ C[s] (6)

is called a polynomial of order n with coefficient vector a = (a0, . . . , an) ∈ Cn+1.

Furthermore, n(a) := deg(p(s, a)) denotes the degree of the polynomial, i.e.

an(a) 6= 0 and aj = 0 for j = n(a) + 1, . . . , n. The coefficient an(a) is called the

leading coefficient.

If the whole unordered n-tuple, i.e. a set in which the multiplicities are taken

into account, of the polynomial p(s, a) is considered, continuous dependence of

these roots on the coefficient vector with respect to the norm defined by (7) can

be established.

The unordered n-tuple of roots of p(s, a) is for any a = (an, . . . , a0) ∈ Cn+1

with an 6= 0 denoted by

Λ(a) = ⌊s1(a), . . . , sn(a)⌋.

Furthermore, if A ∈ Cn×n is a matrix, then Λ(A) denotes the unordered n-tuple

42

of eigenvalues of A, i.e. the roots of the characteristic polynomial χA(s), where

multiplicities are taken into account.

Definition 3.2 (Distance between unordered n-tuples). A distance between

unordered n-tuples is defined by

d(Λ,Λ′) = minπ∈Πn

maxk∈1,...,n

|λπ(k) − λ′k|, (7)

where Λ and Λ′ are unordered n-tuples comprising respectively λ and λ′ and Πn

the group of permutations of the set 1, . . . , n. This metric is called matching

distance.

The corresponding metric space of all unordered complex n-tuples endowed with

the metric (7) is (Tn(C), d).

The following theorem claims that a small perturbation of the coefficient vector

a causes only a small perturbation of the roots and that, if additional roots

occur, their distance from zero is large.

Theorem 3.3. Let p(s, a) be a non-constant polynomial whose degree n(a) is

smaller than or equal to n and which has l distinct roots sj, j ∈ 1, . . . , l, with

corresponding multiplicities mj, j ∈ 1, . . . , l.

Then, for the mutually disjoint closed disks Uε(sj) a δ(ε) > 0 exists, such that

for all coefficient vectors a in the open ball Uδ(ε)(a) there are exactly mj roots

of p(s, a) inside the disk Uε(sj) for j ∈ 1, . . . , l and n(a)− n(a) roots outside

U 1ε(0).

Proof. Step 1: The mj roots associated to sj remain in Uε(sj).

For the proof Rouché’s theorem will be used:

Theorem (Rouché). Let f and g be meromorphic functions on an open and connected

subset G of C. The closed disk Ur(a) with cent re a is contained within G. Furthermore,

f and g do not have poles or zeros on the boundary ∂Ur(a) of Ur(a). It follows from

the inequality

|f(z)− g(z)| < |f(z)|+ |g(z)| ∀z ∈ ∂Ur(a)

that the difference between zeros and poles of the function f equals the the difference

between zeros and poles of g on the whole open disk Ur(a), i.e.

Nf − Pf = Ng − Pg,

where Nf and Pf denote the number of zeros and poles of f in Ur(a).

Proof. See [3].

43

Choose an ε > 0 such that the disks with the distinct zeros as centers are

mutually disjoint, i.e. Uε(sj) ∩ Uε(si) = ∅ ∀i 6= j, and define

µj(ε) = mins∈∂Uε(sj)

|p(s, a)| > 0.

By continuous dependence of the polynomial on the coefficient vector a, a disk

with cent re a and radius δ(µj(ε)) > 0 can be chosen such that every polynomial

p(s, a), a ∈ Uδ(µj(ε))(a), differs only little from the initial polynomial, i.e.

sups∈∂Uε(sj)

|p(s, a)− p(s, a)| < µj(ε) ∀j ∈ 1, . . . , l, ∀a ∈ Uδ(µj(ε))(a)

It follows with Rouché’s theorem that the number of zeros of the two polynomials

p(s, a) and p(s, a) coincide on each Uε(sj) and for every a ∈ Uδ(µj(ε))(a), i.e.

Np(s,a) = Np(s,a) in Uε(sj), ∀a ∈ Uδ(µj(ε))(a).

Step 2: n(a)− n(a) roots are outside of U 1ε(0).

For the second part it is assumed that

• a sequence (ak)k∈N in Uδ(a) exists which converges versus a, i.e. akk→∞−−−−→

a, and that

• for every ak, there exists a root zk of p(·, ak) ∈ U 1ε(0)\

l⋃j=1

Uε(sj).

Since the closure of U 1ε(0)\

l⋃j=1

Uε(sj) is compact, one can choose a subsequence

(zk)k∈N which is convergent in C. But then, the limit z0 ∈ C\l⋃j=1

Uε(sj) would

satisfy p(z0, a) = limk→∞

p(zk, ak) = 0 and hence p(·, a) would have more than

n(a) roots. This contradiction shows that the second assertion also holds.

By the following corollary, the meaning of the expression “the roots depend

continuously on the coefficient vector” is specified.

Corollary 3.4 (Roots depend continuously on coefficient vector). The map

Λ :

x ∈ Cn+1|xn 6= 0 → (Tn(C), d)

a 7→ Λ(a)

associates to every coefficient vector a ∈x ∈ Cn+1 | xn 6= 0

the unordered

n-tuple of the roots of p(s, a). This map is continuous.

44

Proof. This corollary follows directly from theorem 3.3.

Since Cn×n is isomorphic to the space of all linear applications from Cn to Cn,

which is denoted by L(Cn), a linear operator A ∈ L(Cn) is represented as a

matrix in respect to the standard basis of Cn. This space is endowed with an

arbitrarily chosen operator norm ‖·‖ = ‖·‖L(Cn).

Corollary 3.5 (Results for matrices). The map

Λ :

Cn×n → (Tn(C), d)

A 7→ Λ(A)

assigns to every A ∈ Cn×n the unordered n-tuple of roots of its characteristic

polynomial ξA(s) = det(sI −A). This map is continuous on Cn×n, i.e. for any

given A0 ∈ Cn×n and ε > 0 there exists a δ > 0 such that

‖A−A0‖ < δ ⇒ dist(Λ(A),Λ(A0)) < ε,

where the distance between unordered n-tuples is defined by (7).

Proof. This corollary follows directly from theorem 3.3 applied on the charac-

teristic polynomial ξA(s) of A.

This corollary does not mean that the eigenvalues of A ∈ Cn×n can be repre-

sented as continuous functions of A.

The following theorem shows that the roots can be represented as analytic

functions if they are simple.

Theorem 3.6 (Simple root representation). If sj is a root of multiplicity one of

p(s, a), then there exist δ, ε > 0 such that for all coefficients vectors a in Uδ(a)

the following statements hold.

• The polynomial p(s, a) has exactly one root sj(a) in Uε(sj).

• The function sj(a) depends analytically on a in Uδ(a).

• The function sj(a) coincides with sj on the point a, i.e. sj(a) = sj.

45

Proof. This theorem follows from theorem 3.3 and from the residue theorem,

by which1

2πi

∫

Γj

sp′(s, a)

p(s, a)ds = sj(a), a ∈ Uδ(a)

is analytic with respect to a.The prime in the expression p′(s, a) denotes

ddsp(s, a).

The same result can be stated for matrices and their characteristic polynomials.

Corollary 3.7. If λ0 ∈ C is a simple eigenvalue of A0 ∈ Cn×n, then there exist

δ, ε > 0 such that for all matrices A in Uδ(A), the following statements hold.

• All A ∈ Uδ(A0) have exactly one simple eigenvalue λ(A) ∈ Uε(λ0).

• The eigenvalue function λ(A) depends analytically on the entries of A in

Uδ(A0).

• The function λ(A) coincides in A0 with λ0, λ(A0) = λ0

Proof. This corollary follows directly from theorem 3.6 applied to the charac-

teristic polynomial ξA(s).

3.1 Polynomials with Meromorphic Coefficients

In this subsection, the coefficients of a polynomial are members of the field of

meromorphic functionsM(Ω) or members of the ring of holomorphic functions

O(Ω) defined on Ω.

In the following, Ω denotes a domain, i.e. a connected and open set, in the field

of complex numbers. Due to the identity theorem, the ring O(Ω) of analytic

functions on Ω is even an integral domain, i.e. a ring with multiplicative unit

and without zero divisors. Therefore,MΩ is the quotient field of O(Ω).

Furthermore, O(Ω)[s] andM(Ω)[s] are respectively the rings of all the polyno-

mials with coefficients in O(Ω) orM(Ω).

In order to prove the theorems about the polynomials with coefficients in an

arbitrary field, it is necessary to know some facts about these polynomials.

46

Definition 3.8 (Irreducibility). The polynomial p(s, a(z)) ∈ M(Ω)[s] is called

irreducible over M(Ω) if and only if p(s, a(z)) is non-constant and cannot be

represented as a product of two non-constant polynomials, i.e.

• deg(p) ≥ 1

• 6 ∃q ∈M(Ω) : 0 < deg(q) < deg(p) ∧ q|p, where deg(p) denotes the degree

of the polynomial p(s, a(z)).

Definition 3.9 (p(s, a(z)) is prime). p(s, a(z)) ∈ M(Ω)[s] is prime overM(Ω)

if and only if it follows from the fact that p(s, a(z)) divides the product q1(s, b(z))·

q2(s, c(z)) in M(Ω)[s] that p(s, a(z)) divides q1(s, b(z)) or q2(s, c(z)).

Definition 3.10 (p(s, a(z)) and p(s, b(z)) are coprime). Two polynomials

p(s, a(z)), p(s, b(z)) ∈ M(Ω)[s] are called coprime over M(Ω) if and only if

their greatest common divisor is a non-zero constant polynomial.

Theorem 3.11. If p ∈M(Ω)[s] is irreducible, then p is prime.

Definition 3.12 (Resultant matrix of two polynomials). The resultant matrix

R(p, q) ∈ M(Ω)(n+m)×(n+m) of the polynomials p(s, a(z)) =n∑k=0

ak(z)sk and

q(s, b(z)) =m∑k=0

bk(z)sk is defined as

R(p, q) =

an(z) · · · a0(z) 0 · · · 0 0

0 an(z) · · · a0(z). . . 0

.... . .

. . .. . .

. . ....

0. . .

. . .. . . 0

0 0 · · · 0 an(z) · · · a0(z)bm(z) · · · b0(z) 0 · · · 0 0

0 bm(z) · · · b0(z). . . 0

.

... . .

. . .. . .

. . ....

0. . .

. . .. . . 0

0 0 · · · 0 bm(z) · · · b0(z)

.

The coefficient vector (a0(z), . . . , an(z)) ∈ M(Ω)n+1 is shifted (m − 1) times

to the right and the coefficient vector (b0(z), . . . , bm(z)) ∈ M(Ω)m+1 (n − 1)

times to the right. Sometimes the degrees of the polynomials will be indicated

by writing Rn,m(p, q) instead of R(p, q).

• The determinant of Rn,m(p, q) is zero if and only if p and q have a common

factor of positive degree or an = bm = 0.

47

Definition 3.13 (Discriminant matrix of p(s, a(z))). The discriminant ma-

trix of p(s, a(z)) ∈ M(Ω)[s] is defined by Dn(p) = R(p, p′) = Rn,n−1(p, p′) ∈

M(Ω)(2n−1)×(2n−1), where p′(s, a(z)) =n∑k=1

kak(z)sk−1.

• The determinant of the discriminant matrix of a polynomial p(s, a(z)) ∈

M(Ω)[s] is zero if and only if p(s, a(z)) and p′(s, a(z)) have a non-trivial

common factor or if an = 0.

Definition 3.14 (Critical point of a polynomial). Let p(s, a(z)) ∈ M(Ω)[s]

be a polynomial with meromorphic coefficients and with a first coefficient not

identically zero, i.e. an(z) 6≡ 0.

The point z0 ∈ Ω is called a critical point of p(s, a(z)) if and only if one of the

following conditions is satisfied.

• The first coefficient an(z) vanishes at z0.

• The point z0 is a pole of one of the coefficients ai(z) i ∈ 0, . . . , n.

• The polynomial p(s, a(z0)) ∈ C[s] has a strictly smaller number of distinct

roots than p(s, a(z)) for some other z ∈ Ω.

The set of critical points of the polynomial p(s, a(z)) is denoted by Cp.

Definition 3.15 (Critical point of a matrix). If the entries of A(z) depend

analytically on one complex parameter z in a domain Ω ⊆ C, then z0 ∈ Ω is

a critical point of A(·) = (A(z))z∈Ω if and only if z0 is a critical point of the

characteristic polynomial ξA(z)(s) = det(sI −A(z)).

The set of critical points of A(·) is denoted by CA.

48

Now, the first important theorem of this section can be stated. It concludes

about the structure of the critical points and it provides a factorization of the

polynomial p(s, a(z)).

Theorem 3.16 (Critical points are isolated). Let p(s, a(z)) ∈ M(Ω)[s] be a

polynomial with meromorphic coefficients and with a first coefficient an(z) 6≡ 0,

and let D be a simply connected subset of Ω\Cp.

It follows that

• Cp is locally finite in Ω , i.e. K ∩ Cp is finite for compact K ⊆ Ω, that

• there exist n = deg(p) not necessarily distinct analytic functions

s1(z), . . . , sn(z) ∈ O(D), such that

p(s, a(z)) = an(z)n∏

i=1

(s− si(z)) ∀s ∈ D.

and that

• the multiplicity of each root si(z) is constant on D.

Proof. In a first step, the theorem will be proved for irreducible monic polyno-

mials. Subsequently, the theorem will be proved for the general case where

p(s, a(z)) = an(z)l∏i=1

pmii (s, bi(z)) is factorized in irreducible monic factors

pi(s, bi(z)) ∈M(Ω)[s].

Step 1: Proof for irreducible monic polynomials.

Let q(s, b(z)) ∈ M(Ω)[s] be an irreducible monic polynomial of degree m and

let b(z) = (b0(z), . . . , bm−1(z), 1) be its coefficient vector.

Step 1.a: Cq = Zq ∪ Pq is locally finite.

The set Pq of all poles of the coefficients bi(z) of q(s, b(z)) is locally finite because

the poles of a meromorphic function which isn’t identically ∞ do not have a

cluster point in a domain.

The set Zq of multiple roots of q(s, b(z)) is locally finite as well. Since

q(s, b(z)) is irreducible, q(s, b(z)) cannot have a non-constant common fac-

tor with dds (q(s, b(z))) in M(Ω)[s]. Therefore, the discriminant of q(s, b(z)),

49

ψq(z) = Dm(q)(z) ∈ M(Ω), must be a non-identically-zero meromorphic func-

tion on Ω.

ψq(z) is a product of meromorphic functions and therefore meromorphic on

Ω\Pq. For each point z0 ∈ Ω\Pq, ψq(z0) is the discriminant of the complex

polynomial q(s, b(z0)) ∈ C[s]. The aforementioned discriminant is zero in z0 if

and only if q(s, b(z0)) has multiple roots.

Step 1.b: Analytic representation and constant multiplicity.

By theorem 3.6,for every z0 in a simply connected domain D ⊆ Ω\Cq,

there exists a neighborhood U of z0 in D and m distinct analytic functions

s1(·), . . . , sm(·) : U → C such that q(s, b(z)) =m∏i=1

(s − si(z)) and si(z) 6= sj(z)

for i 6= j, z ∈ U . It follows from the monodromy theorem that the representa-

tion of q(s, b(z)) as a product of distinct analytic linear factors can be extended

across the whole simply connected domain D.

Step 2: General case

The above-mentioned result will now be applied to the monic factors of

p(s, a(z)) = an(z)l∏i=1

pmii (s, bi(z)).

Step 2.a: Cp is locally finite.

Step 2.a.1: Pp ∪ Ω0 is locally finite andl⋃i=1

Ppi ⊆ Pp

The set of poles Pp of the coefficients ai(z), i ∈ 0, . . . , n, of the poly-

nomial p(s, a(z)) and the set Ω0 of zeros of the leading coefficient an(z)

are evidently locally finite. Furthermore, it is easy to see that every

pole z0 of a coefficient of pi(s, bi(z)) is a pole of one of the coefficients of

p(s, a(z)) = an(z)l∏i=1

pmii (s, bi(z)). Hence, the set of poles Ppi of the polynomial

p(s, bi(z)) is for every i ∈ 1, . . . , l contained within Pp.

Step 2.a.2: Z =l⋃

i,j=1i6=j

Zi,j is locally finite.

Z =l⋃

i,j=1i6=j

Zi,j denotes the set of common zeros of the polynomials pi and pj

50

for all i 6= j. Since the polynomials pi(s, bi(z)) and pj(s, bj(z)) with respective

degrees ni and nj are coprime for every i, j ∈ 0, . . . , l and i 6= j , the

determinant of the resultant matrix ψi,j(z) = det[Rni,nj (pi, pj)

]∈ M(Ω)

cannot be identically zero on Ω. ψi,j(z) is analytic on Ω\Pp. For each point

z0 ∈ Ω\Pp, ψi,j(z0) is the determinant of the resultant matrix of the complex

polynomials pi(s, bi(z0)) and pj(s, bj(z0)). The function ψi,j(z) is zero for a

z0 ∈ Ω\Pp if and only if pi(s, bi(z0)) and pj(s, bj(z0)) have common roots.

The set Zi,j of zeros of ψi,j(z) in Ω\Pp is locally finite in Ω, and therefore

Z =l⋃

i,j=1i6=j

Zi,j is also locally finite.

Step 2.a.3: Cp = Ω0 ∪ Pp ∪ Z ∪l⋃i=1

Zpi .

If z0 ∈ Ω\(Ω0 ∪ Pp ∪ Z ∪l⋃i=1

Zpi), where Zpi denotes the set of multiple roots

of pi(s, bi(z)), the complex polynomial p(s, a(z0)) has ν =l∑i=1

ni distinct roots.

This is the maximal number of distinct roots of p(s, a(z)) for z ∈ Ω\(Ω0 ∪ Pp).

On the other hand if z0 ∈ Zpi , i.e. pi(s) = pi(s, bi(z0)) has strictly less than ni

distinct roots, or if z0 ∈ Zi,j for some i, j ∈ 1, . . . , l, i 6= j, then p(s, a(z0)) has

strictly less than ν distinct roots. Thus

Cp = Ω0 ∪ Pp ∪ Z ∪l⋃

i=1

Zpi

such that Cp (as a finite union of locally finite sets) is locally finite in Ω.

Step 2.b: Analytic representation and constant multiplicity.

Let D be a simply connected subset of Ω\Cp. Since Cpi = Ppi ∪ Zpi ⊆ Cp

for i ∈ 1, . . . , l, all the irreducible factors pi can be decomposed on D

into linear factors with analytical roots. Hence, there exist n (not neces-

sarily distinct) analytic functions si(·) from D to C such that the equation

p(s, a(z)) = an(z)n∏i=1

(s − si(z)) holds for every s in D. It follows from the

definition of critical points that two roots si(z) and sj(z) of p(s, a(z)) which

coincide at some point z ∈ D ⊆ Ω\Cp coincide across the whole domain D.

Corollary 3.17. If A : Ω→ Cn×n is an analytic matrix function on a domain

Ω ⊆ C it follows that

51

• the set of critical points CA of A(·) CA is locally finite,

• and, if D ⊆ Ω\CA is simply connected, that there exist n analytic functions

λi(·) ∈ O(D), i ∈ 1, . . . , n, such that

Λ(A(z)) = (λ1(z), . . . , λn(z)), ∀z ∈ D and

that the multiplicity of each eigenvalue λi(z) is constant on D.

Proof. This corollary follows directly from theorem 3.16 applied on the charac-

teristic polynomial ξA(s) of A(·).

3.2 Behavior of the roots near critical points

The next theorem describes the behavior of the roots of a polynomial in the

neighborhood of a critical point. The punctured disk Uor (z0) and the cut disk

U−r (x) are defined as

Uor (z0) := Ur(z0)\z0

U−r (x) := z ∈ Uor (x)|0 < arg(z − x) < 2π.

Cycles, which are obtained by analytical continuation of the roots, will play a

crucial role in the remainder of this chapter.

Theorem 3.18 (Cycles). Let p(s, a(z)) ∈ O(Ω)[s] be a monic polynomial,

Ur(z0) ⊆ Ω a disk around the critical point z0 with a sufficiently small ra-

dius r such that Ur(z0)∩Cp = z0 and let l be the number of different roots of

p(s, a(z0)).

The κj different roots of p(s, a(z)), z ∈ Uor (z0), which converge for z → z0

towards sj can be partitioned into hj sets Kj,k, j ∈ 1, . . . , hj, of roots with

the same multiplicity and with respective cardinality qjk on Uor (z0), i.e.

1, . . . , κj = Kj,1, Kj,2, . . . , Kj,hj.

The set sjki(z) | k ∈ 1, . . . , hj, i ∈ 1, . . . , qjk, z ∈ Uoδ (z0) is called the sj-

group of roots of p(s, a(z)).

The qjk-valued function (sjk1(·), . . . , sjkqjk (·)) on Uoδ (z0) obtained by analytic

continuation of a root sjki(z) of p(s, a(z)) along a circular path around z0 in

52

Uor (z0) is called a cycle.

These cycles have the following properties.

• If qjk = 1, the function sjk1(z) can be continued analytically across all of

Ur(z0).

• If qjk ≥ 2, (sjk1(z), . . . , sjkqjk (z)) defines a qjk-valued function on Uor (z0)

with a so called branch point z0 of the order qjk − 1.

• The branches of these functions are represented by a Puiseux series of the

form

sjkm(z) =∞∑

t=0

αjkt

[ei 2πqjkm

(z − z0)1qjk

]t, z ∈ U−r (z0),

where j ∈ 1, . . . , l, k ∈ 1, . . . , hj, l ∈ 1, . . . , qjk and m ∈

1, . . . , qjk.

• All the roots of the sj-group converge for z → z0 towards the root sj of

p(s, a(z0)). sj is called the cent re of the cycle, i.e. limz→z0

sjkl(z) = αjk0 =

sj for j ∈ 1, . . . , l, k ∈ 1, . . . , hj and l ∈ 1, . . . , qjk.

Proof. Step 1: First factorization of the polynomial.

Let s1, . . . sl denote the distinct roots of p(s, a(z0)) . Let m1, . . .ml be the

corresponding multiplicities of s1, . . . , sl. It follows from theorem 3.3 that for

an ε > 0 which renders the closed disks with cent re si mutually disjoint a δ(ε)

in (0, r) can be chosen such that the polynomial p(s, a(z)) for every z ∈ Uδ(ε)(z0)

has the same number of zeros mi in the disk Uε(si) as p(s, a(z0)).

Since Uδ(ε)(z0) ⊆ Ur(z0), it follows that no other critical point than z0 is con-

tained within Uδ(ε)(z0) and that therefore the number of distinct roots κj of

p(s, a(z)) in Uε(sj) is constant for all z ∈ Uoδ(ε)(z0).

On the simply connected cut disk U−r (z0), there exist a total ofl∑j=1

κj ana-

lytic functions si(z), representing the roots of p(s, a(z)). The corresponding

multiplicities of these roots are constant on U−r (z0).

For z ∈ Uoδ(ε)(z0)∩U−r (z0) = U−δ(ε)(z0), si(z) belongs to exactly one of the disks

Uε(sj).

53

Grouping the roots on U−r (z0) accordingly to the various Uε(sj) to which belong

the roots for z ∈ Uδ(ε)(z0) leads to the factorization

p(s, a(z)) =l∏

j=1

κj∏

k=1

(s− sjk(z))µjk , ∀z ∈ U−r (z0).

This factorization satisfies the following points.

• limz→z0

sjk(z) = sj , k ∈ 1, . . . , κj, j ∈ 1, . . . , l

• µjk denotes the constant multiplicity of the root sjk(z) of p(s, a(z)) on

U−r (z0).

•κj∑k=1

µjk = mj.

Step 2: Analytic continuation of the roots sjk.

An arbitrarily chosen sjk(z), where j ∈ 1, . . . , l and k ∈ 1, . . . , κj can be

continued analytically along every arc in Uor (z0)

When the function element sjk(z) in a small neighborhood of z1 = z0 + ρe−iα,

for ρ ∈ (0, r) and α ∈ (0, 2π) fixed, is continued analytically along the circular

arc

γ(t) = z0 + ρeit, t ∈ [−α, 2π − α],

a new function element sjk(z) at z1 is obtained.

Step 2.a: The factorization is preserved under analytic continua-

tion.

Due to the identity theorem, the factorization of p(s, a(z)) =l∏

j=1

κj∏

k=1

(s − sjk(z))µjk is preserved by analytic continuation. This is be-

cause p(s, a(z)) = ˜p(s, a(z)) =l∏

j=1

κj∏

k=1

(s− sjk(z))µjk for all z ∈ U−r (z0).

Step 2.b: The roots remain in Uε(sj).

When sjk(z) is continued at some point in U−δ (z0) along a circular arc in Uoδ (z0),

the resulting root element sjk(z) remains in Uε(sj).

• The resulting root element sjk(z) coincides with one of the analytic roots

sjk′ (z), k′ ∈ 1, . . . , κj.

54

• Since multiplicities remain constant on the simply connected set U−r (z0) it

follows that µjk = µjk′ = µjk. Therefore, the multiplicity of the resulting

root element sjk(z) is the same one as that of the analytically continued

root sjk(z).

Step 3: Second factorization.

When the application πz0 : sjk(·) 7→ sjk′ (·) denotes analytic continuation along

a circular arc around z0, not only the roots associated to one circle Uε(sj) remain

invariant but also the set of roots with identical multiplicity.

Therefore, the set 1, . . . , κj can be partitioned into Kj,1, . . . , Kj,hj. hj

is the number of different multiplicities. The sets Kj,i, i ∈ 1, . . . , hj, contain

the indexes of roots in Uε(sj) with identical multiplicity.

The factorization of p(s, a(z)), taking those invariance groups into account, is

p(s, a(z)) =l∏

j=1

hj∏

k=1

qjk∏

i=1

(s− sjki(z))µjk , z ∈ U−r (z0),

where qjk is the cardinality of Kj,k, i.e. qjk = |Kj,k|.

For every j ∈ 1, . . . , l and k ∈ 1, . . . , hj sjkm(z) is analytically continued

to sjk(m+1)(z), m ∈ 1, . . . , qjk − 1 and sjkqjk (z) is analytically continued to

sjk1(z) on Uor (z0).

Step 4: Representation of cycles by Puiseux series

The qjk-valued function f(z) which belongs to the partition Kj,k and to the

cycle (sjk1(z), . . . , sjkqjk (z)) associates with every point z in the punctured disk

Uor (z0) a tuple of qjk distinct values of the following form.

f(z) =

(sjk1(z), . . . , sjkqjk (z)

), if z ∈ U−r (z0)

(limtր2π

sjki(z0 + ρeit)|i ∈ 1, . . . , qjk

), if z = z0 + ρ, 0 < ρ < r

By continuity, sj = limz→z0

sjk(z) holds for every k ∈ 1, . . . , κj. This limit is

called cent re of the cycle.

• If qjk = 1, the point z = z0 is a removable singularity of f(z) such that

f(z) defines an analytic function on the whole disk Ur(z0).

55

• If qjk ≥ 2, z0 is called a branch point of order (qjk−1). The root functions

sjkm(z),m ∈ 1, . . . , qjk are called the branches of f(z) on U−r (z0).

Step 4.a: Construction of the series.

Firstly, set ζ = (z − z0)1qjk and define two sets

Uro(0) = ζ ∈ C : 0 < |ζ| < r

1qjk and

Cm = s ∈ Uro(0) : m−1

qjk2π ≤ arg(s) < m

qjk2π, k ∈ 1, . . . , qjk.

It follows that the application

ϕ :

Uro(0) → Uor (z0)

ζ 7→ z0 + ζqjk

maps every sector Cm bijectively on Uor (z0).

The point ζ1 in the sector C1 is now considered. The point ζ1 is in the pre-image

ϕ−1(z1) = ζn = ρ1q e−

iαq

+n−1q

2π : n ∈ 1, . . . , qjk

of a point z1 = z0 + ρe−iα, which in turn is in the cut disk U−r (z0). The map

ζϕ−→ ϕ(ζ)

sjk1−−−→ sjk1(ϕ(ζ))

is analytic near ζ1. It determines a power series S1(ζ) at ζ1. The continuation

of S1(ζ) along all paths in Uro(0) is noted F (ζ).

Step 4.b: F (ζ) is single-valued.

As ζ travels along a circular arc γ in the ζ - plane once around 0, ϕ(ζ) = z0 + ζq

travels q times around z0 in Uor (z0). Therefore, S1(ζ) = sjk1(ϕ(ζ)) is an-

alytically continued via S2(ζ) = sjk2(ϕ(ζ))), . . . , Sqjk (ζ) = sjkqjk (ϕ(ζ)), to

Sqjk+1(ζ) = S1(ζ). Hence, S1(ζ) = sjk1(ϕ(ζ)) remains unchanged by analytic

continuation along γ and F (ζ) coincides with sjki(z) on the open sector int(Ci).

Step 4.c: Laurent series expansion.

The Laurent series expansion of F (ζ) is a priori

F (ζ) =∞∑

k=−∞

αkζk on Ur

o(0).

However, since limζ→0

F (ζ) = α0 = sj, it follows that all coefficients αk are zero

for k < 0.

56

Thus,

F (ζ) = sj +∞∑

k=1

αkζk, ζ ∈ U

r1q

(0)

is analytic and satisfies

F (ζ) = sjkm(ϕ(ζ)), ζ ∈ int(Cm), m ∈ 1, . . . , qjk.

The qjk values of f at z ∈ Uor (z0) are given by F (ζν), ν ∈ 1, . . . , q, where ζν

runs through all the qjk-th roots of (z − z0), i.e.

f(z) =F (ζν) | ζν = ρ

1qjk e

i νqjk

2π+i θqjk ∧ ν ∈ 1, . . . , qjk

,

where z = z0 + ρeiθ ∈ Uor (z0) and 0 ≤ θ < 2π.

The same result can also be established for the characteristic polynomial of an

analytic matrix function A(z).

Corollary 3.19. Let A : Ω→ Cn×n be an analytic matrix function on a domain

Ω ⊆ C. Let Ur(z0) ⊆ Ω be a disk around the critical point z0 ∈ CA with a

sufficiently small radius r such that Ur(z0)∩CA = z0 and let l be the number

of different eigenvalues of A(z).

The κj different eigenvalues of A(z), z ∈ Uor (z0), which converge for z → z0

towards λj , can be partitioned into hj sets of roots with the same multiplicity

and with respective cardinality qjk on Uor (z0), i.e.

1, . . . , κj = Kj,1, Kj,2, . . . , Kj,hj.

The set λjki(z) | k ∈ 1, . . . , hj, i ∈ 1, . . . , qjk, z ∈ Uoδ (z0) is called the λj-

group of eigenvalues of A(z).

The qjk-valued function (λjk1(·), . . . , λjkqjk (·)) on Uoδ (z0) obtained by analytic

continuation of a root λjki(z) of p(s, a(z)) along a circular path around z0 in

Uor (z0) is called a cycle.

These cycles have the following properties.

• If qjk = 1, the function λjk1(z) can be continued analytically across

Ur(z0).

57

• If qjk ≥ 2, (λjk1(z), . . . , λjkqjk (z)) defines a qjk-valued function on Uor (z0)

with a so called branch point z0 of the order qjk − 1.

• The branches of these functions are represented by a Puiseux series of the

form

λjkm(z) =∞∑

t=0

αjkt

[ei 2πqjkm

(z − z0)1qjk

]t, z ∈ U−r (z0),

where j ∈ 1, . . . , l, k ∈ 1, . . . , hj and m ∈ 1, . . . , qjk.

• All the different eigenvalues in the λj-group converge for z → z0 to-

wards the eigenvalue λj of A(z0). λj is called the cent re of the cycle,

i.e. limz→z0

λjkl(z) = αjk0 = λj for j ∈ 1, . . . , l, k ∈ 1, . . . , hj and

m ∈ 1, . . . , qjk.

Proof. This corollary follows directly from theorem 3.18 applied on the charac-

teristic polynomial of A(·).

3.3 Continuous representation of the root functions

If z0 ∈ Cp is a branch point for the roots of p, that is that the permutation πz0

defined by the analytic continuation of the roots along a circular path around z0

is not the identity, it is not possible to find continuous (single-valued) functions

si(z), i ∈ 1, . . . , n representing the complete set of roots p(s, a(z)) on a disk

around the branch point z0. However, if the parameter z is restricted to a real

interval, it is possible to find a continuous parametrization of the roots. In

this case, it is sufficient to assume the continuous dependence of the coefficient

vector on z = r in order to prove the continuous dependence of the roots. The

following theorem is taken from [11].

Theorem 3.20 (Real case). If I ⊆ R is an interval in R and p(s, a(r)) ∈

C(I,C)[s] is a monic polynomial of degree n, then there exist n continuous func-

tions si : I → C such that

Λ(a(r)) = ⌊s1(r), . . . , sn(r)⌋, ∀r ∈ I.

Proof. A subinterval I0 of I has the property (A) if there exist n continuous

functions on I0 representing the roots of the polynomial p(s, a(r)).

58

Step 1: Property (A) is local.

Let ⌊s(1)1 (r), . . . , s(1)

n (r)⌋ and ⌊s(2)1 (r), . . . , s(2)

n (r)⌋ be representations of Λ(a(r))

in I1 and I2 respectively. It is assumed that

• I1 6⊆ I2 ∧ I2 6⊆ I1 and

• I1 lies to the left of I2.

For r0 ∈ I1∩I2 and after suitable renumbering s(1)i (r0) = s

(2)i (r0), i ∈ 1, . . . , n

it follows that ⌊s(1)1 (r0), . . . , s(1)

n (r0)⌋ and ⌊s(2)1 (r0), . . . , s(2)

n (r0)⌋ represent the

same Λ(a(r0)).

The functions s(0)i (r), i ∈ 1, . . . , n, are defined on I0 by

s(0)i (r) =

s

(1)i (r), r ≤ r0

s(2)i (r), r ≥ r0.

These functions are continuous and represent Λ(a(r)) on I0.

If every point r in a subinterval I ′ of I has a neighborhood U with property

(A), then the whole subinterval I ′ has property (A). The property (A) is local.

Step 2: Proof by an induction argument.

For the case n = 1, the claimed representation is evident. To conclude in the

case (n − 1) 7→ n, it will be shown that, for every point in a open subset of I,

there exists a neighborhood which has property (A). This relation is equivalent

to the fact that the open subset has property (A).

The set

Θ = r ∈ I | n elements of Λ(a(r)) are identical

is a pre-image of a singleton and therefore closed in I. Its complement

Ξ = I\Θ

is open in I.

Step 2.a: Every r in Ξ has a neighborhood U with property (A)

Since n elements of Λ(a(r0)), r0 ∈ Ξ, are not all identical, they can be divided

into two separate groups with n1 and n2 elements, where n1 +n2 = n. In other

59

words, Λ(a(r0)) is composed of an n1-tuple and an n2-tuple. The elements in

the n1-tuple are different to the elements in the n2-tuple; there is no element in

the n1-tuple of the same value as an element in the n2-tuple.

Step 2.b: Λ(a(r)) is continuous on Uε(r0).

Since Λ(a(r)) consists of an n1-tuple an n2-tuple for sufficiently small |r − r0|,

it follows on from the induction hypothesis that these tuples can be represented

by families of continuous functions in a neighborhood Uε(r0) of r0.

The combination of the two tuples represents Λ(a(r)) in Uε(r0). Hence, Uε(r0)

has property (A).

Step 2.c: Representation of Λ(a(r)).

Since Ξ is open in I, I consists, at most, of a countable number of subintervals

I1, I2, . . .. Since every r ∈ Ξ has a neighborhood Uε(r) with property (A), each

component interval Ip has property (A).

The n continuous functions si(r), i ∈ 1, . . . , n, which represent Λ(a(r)) on I

are defined by

si(r) =

s

(p)i (r), r ∈ Ip ∧ p ∈ 1, 2, . . .

s(r) r ∈ Θ.

Again, the theorem can be established for matrices and the corresponding eigen-

values too.

Corollary 3.21 (Real case). If I ⊆ R is an interval in R and A(τ) ∈ Cn×n

depends continuously on τ in I, then there exist n continuous functions λi : I →

C such that

Λ(A(τ)) = ⌊λ1(τ), . . . , λn(τ)⌋ ∀τ ∈ I.

Proof. Apply theorem 3.20 to the characteristic polynomial ξA(τ)(s) = det(sI−

A(τ)).

If A(τ) is differentiable in τ on an interval I ⊆ R, and ifA(τ) is diagonalizable for

all τ ∈ I, then, by a theorem of Kato, the λi(·) can be chosen to be differentiable.

60

4 Smoothness of Eigenprojections and Eigen-

vectors

In this section, the behavior of eigenprojections and eigenvectors for the case

of not necessarily distinct eigenvalues will be investigated. It will be shown

that the eigenprojections have necessarily poles on branch points. Furthermore,

the normality condition, which prohibits the existence of any two root functions

with the same multiplicity (i.e. branch points can not occur), will be introduced.

In particular, this condition is satisfied by an self-adjoint operator.

4.1 Some facts about resolvents and eigenprojections of

matrices

Definition 4.1 (Resolvent of A at s). Let A be a matrix in Cn×n and ρ(A) =

C\σ(A) the resolvent set of A. Then

R(s,A) = (sI −A)−1, s ∈ ρ(A)

is called the resolvent of A at s ∈ ρ(A).

Some facts about resolvents will now be derived.

Lemma 4.2.

• R(s,A) commutes with A.

• The resolvent equation

R(s,A)−R(s0, A) = (s0 − s)R(s,A)R(s0, A)

holds for all s, s0 in ρ(A).

• R(s,A) and R(s0, A) commute.

• R(s,A) admits the absolutely convergent power series expansion

R(s,A) =∞∑

k=0

(s− s0)k(R(s0, A))k+1, |s− s0| < ‖R(s0, A)‖−1

61

for every s0 ∈ ρ(A). Therefore, R(s,A) is analytic on ρ(A) and its deriva-

tives at s0 ∈ ρ(A) are

R(k)(s0, A) = k!(R(s0, A))k+1, k ∈ N∗.

Proof. Step 1: R(s,A) commutes with A. Since the two equations

(sI −A)−1(sI −A) = s(sI −A)−1 − (sI −A)−1A = I

(sI −A)(sI −A)−1 = s(sI −A)−1 −A(sI −A)−1 = I

hold, it follows that

A(sI −A)−1 = (sI −A)−1A.

Step 2: Resolvent equation.

If s 6= s0, it follows that

R(s,A)−R(s0, A) = R(s,A)R(s0, A)(s0I − A)︸ ︷︷ ︸=I

− (sI −A)R(s,A)︸ ︷︷ ︸=I

R(s0, A)

= −R(s,A)R(s0, A)A+AR(s,A)R(s0, A)︸ ︷︷ ︸=0,

( because A and R(s,A) commute )

+(s0 − s)R(s,A)R(s0, A).

Step 3: R(s,A) and R(s0, A) commute.

Together with the resolvent equation, it follows from

R(s,A)−R(s0, A) = (s0I −A)R(s0, A)︸ ︷︷ ︸=I

R(s,A)−R(s0, A)R(s,A)(sI −A)︸ ︷︷ ︸=I

= −AR(s0, A)R(s,A) +R(s0, A)R(s,A)A︸ ︷︷ ︸=0,

( because A and R(s,A) commute )

+(s0 − s)R(s0, A)R(s,A),

that R(s,A) and R(s0, A) commute.

Step 4: Power series representation.

The resolvent equation is equivalent to

R(s0, A) = R(s,A) + (s0 − s)R(s,A)R(s0, A)

= R(s,A)[I − (s− s0)R(s0, A)], s, s0 ∈ ρ(A).

Therefore, the following absolutely convergent series expansion of R(s,A) at

s0 ∈ ρ(A)

R(s,A) =∞∑

k=0

(s− s0)k(R(s0, A))k+1, |s− s0| < ‖R(s0, A)‖−1

62

is obtained.

Thus, R(s,A) is analytic on ρ(A) and its derivatives at s0 ∈ ρ(A) are

R(k)(s0, A) = k!(R(s0, A))k+1, k ∈ N∗.

In the following, λ1, . . . λl denote the distinct eigenvalues of A ∈ Cn×n and

m1, . . . ,ml their corresponding multiplicities.

Cn can be decomposed into A-invariant generalized eigenspaces.

Cn = ker(λ1I −A)m1 ⊕ · · · ⊕ ker(λlI −A)ml .

Definition 4.3 (Eigenprojection Pj of A). The map

Pj :

Cn → ker(λjI −A)mj

x1 ⊕ · · · ⊕ xl 7→ xj, j ∈ 1, . . . , l

is called the eigenprojection of A for the eigenvalue λj.

Definition 4.4 (Eigennilpotent Nj of A). The operator

Nj = (A− λjI)Pj : Cn → Cn

is called the eigennilpotent of A for the eigenvalue λj .

Properties

(i).l∑j=1

Pj = I, PjPk = δjkPj , ⊲⊳∈ 1, . . . , l

(ii). PjA = APj = λjPj + (A− λjI)Pj = λjPj +Nj

(iii). Al∑j=1

Pj =l∑j=1

(λjPj +Nj)

(iv). A is diagonalizable if and only if Nj = 0, j ∈ 1, . . . , l.

(v). If A is diagonalizable, i.e. A =∑lj=1 λjPj , it follows that the resolvent

can be written as R(s,A) =∑lj=1(s− λj)−1Pj for every s ∈ ρ(A)

(vi). If A is normal, then A is diagonalizable. It also follows that the eigen-

projections Pj are self-adjoint (i.e. Pj = P ∗j , j ∈ 1, . . . , l) and that the

spectral norm of Pj is 1 (i.e. the biggest eigenvalue is 1).

63

(vii). If A is real and symmetric, then the eigenvalues λj , j ∈ 1, . . . , l, are real

and the eigenprojections are real and symmetric.

4.2 Integral representation of the eigenprojections

Lemma 4.5 (Partial fraction expansion of R(s,A)). Let λ1, . . . λl be the l dis-

tinct eigenvalues of A ∈ Cn×n. Let m1, . . . ,ml be the corresponding multiplici-

ties. Let P1, . . . , Pl be the associated eigenprojections. And let N1, . . . , Nl be the

associated eigennilpotents.

It then follows that R(s,A) can be

R(s,A) =l∑

j=1

[Pj

(s− λj)+mj−1∑

k=1

Nkj(s− λj)k+1

]. (8)

If Γj is a positively oriented circle in ρ(A) enclosing exclusively λj , it follows

that the eigenprojection can be written as the integral

Pj =1

2πi

∫

Γj

R(s,A)ds. (9)

Proof. Step 1: Without loss of generality, A has Jordan canonical

form.

It can be easily seen that if A = T−1AT , where T ∈ GLn(C) is a non singular

matrix, the equations

R(s, A) = T−1R(s,A)T, Pj = T−1PjT, and Nj = T−1NjT,

hold as well for every j in 1, . . . , l.

Therefore, A can be written as

A =l⊕

j=1

rj⊕

k=1

J(λj , njk),

where J(λj , njk) is defined as

J(λj , njk) =

λj 1 0 · · · 0

0 λj 1. . .

......

. . .. . .

. . . 0...

. . . λj 1

0 · · · · · · 0 λj

, ∈ Cnjk×njk , k ∈ 1, . . . , rj.

64

In the above expression, rj is the geometric multiplicity of the eigenvalue λj , njk

the dimension of the k-th Jordan block of the eigenvalue λj and mj =rj∑k=1

njk

the algebraic multiplicity of the eigenvalue λj .

The associated eigenprojections Pj and eigennilpotents are,

Pj = 0m1 ⊕ · · · ⊕ 0mj−1 ⊕ Imj ⊕ 0mj+1 ⊕ · · · ⊕ 0ml

Nj = 0m1 ⊕ · · · ⊕ 0mj−1 ⊕rj∑k=1

J(0, njk)⊕ 0mj+1 ⊕ · · · ⊕ 0ml ,

where In denotes an identity matrix of dimension n and 0n a matrix of zeros

of dimension (n× n).

Step 2: Formula for R(s,A).

Since A has Jordan normal form, the resolvent can be written as

R(s,A) =l⊕

j=1

rj⊕

k=1

R(s, J(λj , njk)).

Furthermore, for any λ ∈ C, m ∈ N,

R(s, J(λ,m)) =

(s− λ) −1 0. . .

. . .

. . . −1

0 (s− λ)

−1

=

=1

det(R(s, J(λ,m)))

(R(s, J(λ,m))

)=

=

(s− λ)−1 (s− λ)−2 · · · (s− λ)−m

0. . .

. . ....

.... . .

. . . (s− λ)−2

0 · · · 0 (s− λ)−1

= (s− λ)−1Im +m−1∑

k=1

(s− λ)−k−1Nk.

Step 3: Integral representation of Pj.

By Cauchy’s formula,

1

2πi

∫

γ

f(s)

(s− λ)k+1ds =

f (k)(s)

k!

∣∣∣∣s=λ

.

65

In the above expression, γ is a positively oriented circle around λ and f(z) an

analytic function. It follows that

1

2πi

∫

Γj

R(s, J(λi, nik))ds =

0nik if i 6= j

1nik if i = j

Since

Pj = 0m1 ⊕ · · · ⊕ 0mj−1 ⊕ 1mj ⊕ 0mj+1 ⊕ · · · ⊕ 0ml ,

this proves the assertion.

Corollary 4.6. Let λ1, . . . λl denote the l distinct eigenvalues of A ∈ Cn×n

and let Γ be a positively oriented simple closed curve in ρ(A), enclosing the

eigenvalues of A. It follows that the equation

1

2πi

∫

Γ

R(s,A)ds =l∑

j=1

Pj

holds.

Proof. Due to the residue theorem, it follows that

1

2πi

∫

Γ

R(s,A)ds =l∑

j=1

Res(R(s,A), λj) =l∑

j=1

1

2πi

∫

Γj

R(s,A)ds =l∑

j=1

Pj ,

where Γj , j ∈ 1, . . . , k are small circles around λj . Thus, the assertion follows

from corollary 4.6.

4.3 Analyticity at non-critical points

In this section, the analyticity of the eigenprojections and the eigenbasis is inves-

tigated. Firstly, the total projection will be defined in order to understand the

behavior of the projections when small changes occur. Secondly, the obtained

results will be applied to the case where the matrices depend analytically on a

complex parameter. Lastly, the analytical dependence of the eigenvectors will

be proved.

66

4.3.1 Analyticity with respect to matrices

Definition 4.7 (λj - group of eigenvalues of A = A0 +). Let λ1, . . . λl be the

distinct eigenvalues of A0 ∈ Cn×n and let m1, . . . ,ml be the corresponding mul-

tiplicities. Furthermore, Γj, j ∈ 1, . . . , l, denote the non-overlapping circles

around each λj such that for all ∈ Cn×n with a sufficiently small norm, Γj

encloses exactly mj eigenvalues of A = A0 +, taking account of multiplicities.

The set (or the unordered mj-tuple) of the eigenvalues of A = A0 + is called

λj - group of A.

Definition 4.8 (Total projection of the λj - group). Under the same framework

as the preceding definition, the total projection for the λj-group of eigenvalues

of A = A0 +,

P totalj (A) =1

2πi

∫

Γj

R(s,A)ds, ‖A−A0‖ < mins∈Γj‖R(s,A0)‖−1.

• It follows from corollary 4.6 that the total projection coincides with the

sum of eigenprojections of all the eigenvalues of A lying inside Γj .

• Pj(A0) coincides with the eigenprojection for the eigenvalue λj of A0.

Proposition 4.9. Let λ1, . . . λl be the distinct eigenvalues of A0 ∈ Cn×n. Let

m1, . . . ,ml be the corresponding multiplicities. Let ‖A − A0‖ be smaller than

‖R(s,A0)‖−1. Then, it follows that the total projections Pj(A) depend analyti-

cally on the entries of A.

Proof. Step 1: Power series representation of R(s,A).

∈ Cn×n is defined by A − A0 = , where ‖‖ < mins∈Γj‖R(s,A0)‖−1 for all

j ∈ 1, . . . , l. Since

R(s,A) = (sI − (A0 +))−1 =

= [(sI −A0)− (sI −A0)−1(sI −A0)︸ ︷︷ ︸=I

]−1 =

= [(I − (sI −A0)−1

︸ ︷︷ ︸=R(s,A0)

)(sI −A0)]−1 =

= (sI −A0)−1[I −R(s,A0)]−1,

67

it follows that the power series

R(s,A) = R(s,A0)∞∑

k=0

[(A −A0)R(s,A0)]k

converges uniformly for s ∈ Γj and for every j ∈ 1, . . . , l.

Step 2: Power series representation of Pj(A) in the entries of (A−

A0).

With the power series representation of R(s,A), the formula

Pj(A) =1

2πi

∞∑

k=0

∫

Γj

R(s,A0)[(A −A0)R(s,A0)]kds

is obtained easily.

4.3.2 Analyticity with respect to a parameter

If ‖A−A0‖ is sufficiently small then A has at least as many distinct eigenvalues

as A0. On the other hand, every neighborhood of A0 ∈ Cn×n contains matrices

with any number of different eigenvalues between l = |σ(A0)| and n. In the

case of matrices depending analytically on a complex parameter, the situation

greatly simplifies.

The following theorem, taken from [11], permits to define the eigenprojections

on a simply connected domain using Kato’s construction of globally defined

transformation matrices U(z, z0).

Theorem 4.10 (Global definition of the transformation functions). Let

P1(z), . . . , Pn(z) be analytic projection-valued functions on a simply connected

subset D ⊆ C and define Q(z) = 12

l∑k=1

[P ′k(z)Pk(z)− Pk(z)P ′k(z)]. The projec-

tions satisfy the following conditions.

(i).l∑j=1

Pj(z) = In

(ii). Pj(z)Pk(z) = δjkPk(z) ∀j, k ∈ 1, . . . , n

(iii). For every z0 in D, U(·, z0) is the unique solution of the operator differential

equation X ′(z) = Q(z)X(z), X(z0) = I.

It then follows that the unique solution has the following properties.

68

(i). U(z0, z0) = I

(ii). U(z1, z)U(z, z0) = U(z1, z0)

(iii). U(z, z0)−1 = U(z0, z)

(iv). Pj(z) and Pj(z0) are similar, i.e. Pj(z)U(z, z0) = U(z, z0)Pj(z0).

(v). If D is symmetric with respect to the real axis, i.e. when z is an element of

D then z is contained within D as well, and if the matrices P (z) and P (z0)

are self-adjoint, i.e. P (z)∗ = P (z), P (z0)∗ = P (z0), then U(z, z0)∗ =

U(z, z0)−1.

In particular, if z, z0 ∈ D are real and P (z) and P (z0) are hermitian (resp.

real and symmetric), then U(z, z0) is unitary (resp. orthogonal)

Proof. By differentiating P 2(z) = P (z) and Pj(z)Pk(z) = δjkPk(z) it follows

that

• P ′j(z)Pj(z) + Pj(z)P ′j(z) = P ′j(z), j ∈ 1, . . . , n, and

• Pj(z)P ′k(z) = P ′j(z)Pk(z), k 6= j.

Step 1: P ′j(z) = Q(z)Pj(z)− Pj(z)Q(z).

Q(z)Pj(z) = 12

l∑k=1

P ′k(z)Pk(z)Pj(z)︸ ︷︷ ︸

=δjkPj(z)

− Pk(z)P ′k(z)︸ ︷︷ ︸P ′k

(z)−P ′k

(z)Pk(z)

Pj(z)

=

= 12P′j(z)Pj(z) + 1

2

l∑k=1

P ′k(z)Pk(z)Pj(z)︸ ︷︷ ︸

δjkPj(z)

−P ′k(z)Pj(z)

=

= P ′j(z)Pj(z)− 12

l∑k=1

P ′k(z)Pj(z)︸ ︷︷ ︸=

−Pk(z)P ′

j(z) k 6= j

−Pj(z)P ′j

(z) + P ′j

(z) else

=

= P ′j(z)Pj(z) + 12

(l∑

k=1

Pk(z)

)

︸ ︷︷ ︸=I

P ′j(z)

− P ′j(z)

=

= P′j(z)Pj(z)

69

In the same manner one can conclude that Pj(z)Q(z) = −Pj(z)P ′j(z). From

these two equations it follows that

P ′j(z) = (Pj(z)2)′ = P ′j(z)Pj(z) + Pj(z)P ′j(z) = Q(z)Pj(z)− Pj(z)Q(z)

Step 2: Properties of the transformation function.

The first two claims about the evolution operator U(·, z0) are known from the

theory of operator differential equation. The third one is a consequence of the

second by setting z1 = z0.

For the last equation, the following differential equations are considered.

(Pj(z)U(z, z0))′=P ′j(z)U(z, z0) + Pj(z) U ′(z, z0)︸ ︷︷ ︸=Q(z)U(z,z0)

=(P ′j(z) + Pj(z)Q(z))U(z, z0)

=Q(z)Pj(z)U(z, z0)

(U(z, z0)Pj(z0))′=U ′(z, z0)Pj(z0)

=Q(z)U(z, z0)P (z0)

Since the solution of the equation X ′(z) = Q(z)X(z) is unique, the equation

Pj(z)U(z, z0) = U(z, z0)Pj(z0) must hold for every j ∈ 1, . . . , n. The initial

value of this solution is X(z0) = Pj(z0).

Step 3: If P (z) and P (z0) are self-adjoint, it follows that

U(z, z0)−1 = U(z, z0)∗.

Since the equations

Q(z)∗ = 12

l∑k=1

[Pk(z)∗P ′k(z)∗ − P ′k(z)∗Pk(z)∗]

= 12

l∑k=1

[Pk(z)P ′k(z)− P ′k(z)Pk(z)]

= −Q(z)

andd

dzU(z, z0)∗ = −U(z, z0)∗Q(z)

hold, it follows that

ddz (U(z, z0)∗U(z, z0)) = U ′(z, z0)∗U(z, z0) + U(z, z0)∗U ′(z, z0) =

= −U(z, z0)∗Q(z)U(z, z0) + U(z, z0)∗Q(z)U(z, z0) = 0.

Therefore, the product U(z, z0)∗U(z, z0) must be constant.

70

The initial conditions, for both differential equations at z = z0 are U(z0, z0) = I

and U(z0, z0) = I respectively. In view of this fact, it follows that

U(z, z0)∗U(z, z0) = I ∀z ∈ D.

Given that the dimension of the underlying space is finite, this is equivalent to

U(z, z0)∗ = U(z, z0)−1.

Corollary 4.11 (Analyticity of total projections). Let A : Ω → Cn×n be

analytic on the domain Ω ⊆ C, let λ1, . . . , λl be the l distinct eigenvalues of

A0 = A(z0) and let m1, . . . ,ml be the corresponding multiplicities. If |z − z0| is

sufficiently small, then the total projection Pj(z) := Pj(A(z)) depends analyti-

cally on z and has constant rank, rank(Pj(z)) = mj , j ∈ 1, . . . , l.

Proof. Since the total projection has a power series representation in the entries

of A(z) it depends analytically on z on a small disk. The rank is constant

because Pj(z) and Pj(z0) are similar due to theorem 4.10.

Moreover, the total projection coincides on non critical points with the eigen-

projection. These are analytic on any simply connected subset of Ω which does

not contain critical points.

Corollary 4.12 (Analyticity of eigenprojections). Let A : Ω → Cn×n be an-

alytic on the domain Ω ⊆ C and λj : D → C, j ∈ 1, . . . , l, be the l dis-

tinct eigenvalue functions on the simply connected set D ⊆ Ω\CA such that

σ(A(z)) = λ1(z), . . . , λl(z) for z ∈ D.

Then, it follows that

• the corresponding eigenprojections Pj(z), j ∈ 1, . . . , l are analytic and

have constant rank, rank(Pj(z)) = mj , j ∈ 1, . . . , l, on D, that

• the eigennilpotents Nj = (A(z)− λj(z)I)Pj(z), j ∈ 1, . . . , l are analytic

on D as well and that

71

• A(z) admits the spectral representation

A(z) =l∑

j=1

(λj(z)Pj(z) +Nj(z)), ∀z ∈ D.

Proof. Step 1: Pj(z) has constant rank on D ⊆ Ω\CA.

The claim follows directly from theorem 4.10, because Pj(z) and Pj(z0) are

similar for z, z0 ∈ D.

Step 2: The eigenprojection are analytic on D ⊆ Ω\CA.

Let A0 = A(z0) be the matrix belonging to an arbitrary non-critical point z0 ∈

Ω\CA. Let Γj , j ∈ 1, . . . , l, be non-overlapping circles around the l = l(z0)

eigenvalues of A0.

For every z0 ∈ D exists a disk with cent re z0 such that

• there is exactly one eigenvalue λj(z) of A(z) inside the circle Γj and

• the total projection Pj(z) =1

2πi

∫

Γj

R(s,A(z))ds = Res(R(s,A(z)), λj(z))

is the eigenprojection for λj(z).

Therefore, the analyticity of the eigenprojections follows from Proposition 4.9.

4.3.3 Analyticity and Construction of the generalized eigenbasis

Given an analytic eigenvalue λj(·) of algebraic multiplicity mj on a subdomain

D of Ω\CA, it is of interest to find for each z ∈ D mj generalized eigenvec-

tors vj,k(z), k ∈ 1, . . . ,mj. They form a basis for the generalized eigenspace

ker(λj(z)I −A(z))mj and they depend analytically on z ∈ D.

Corollary 4.13. Let P : D → Cn×n be an analytic projection-valued func-

tion on a simply connected set D ⊆ C, let z and z0 be points in D and let

(v1, . . . , vm) be a basis for Im(P (z0)) = P (z0)Cn. Furthermore, let U(z, z0) be

the transformation function from theorem 4.10.

Under these assumptions, the vectors vi(z) = U(z, z0)vi, i ∈ 1, . . . ,m, form a

basis of Im(P (z)) ⊆ Kn which depends analytically on z ∈ D.

72

Proof. Since

P(z)vi(z) = P (z)U(z, z0)vi = U(z, z0)P (z)vi = U(z, z0)vi = vi(z)

the vectors vi(z), i ∈ 1, . . . ,m belong to Im(P (z)). Since dim(Im(P (z))) =

dim(Im(P (z0))) and since U(z, z0) is invertible, the vectors vi(z), i ∈ 1, . . . ,m,

are linearly independent and form for every z ∈ D a basis of Im(P (z)).

Corollary 4.14. Let A : Ω → Cn×n be analytic on the domain Ω ⊆ C. Fur-

thermore, λj is an eigenvalue of algebraic multiplicity mj of A(z0), z0 ∈ Ω\CA.

It follows that there exists

• an analytic function λj(·) on Uδ(z0) representing an eigenvalue of alge-

braic multiplicity mj of A(z).λj(·) coincides in z0 with λj , i.e. λj(z0) =

λj.

• Furthermore, there are mj analytic vector functions (vj,1(z), . . . , vj,mj (z))

from Uδ(z0) to Cn constituting a basis of the generalized eigenspace

ker(λj(z))I −A(z))mj of A(z) for all z in Uδ(z0).

Proof. Step 1: λj(z) is analytic eigenvalue function for A(z).

Since z0 is a non-critical point, the λj - group of eigenvalues of A(z) near

z = z0 consists of only one eigenvalue λj(z) of multiplicity mj . This eigenvalue

of A(z) depends analytically on z in a small disk Uδ(z0).

Step 2: (vj,1(z), . . . , vj,mj (z)) is a basis of the generalized eigenspace.

The associated total projection Pj(z) coincides with the eigenprojection for

λj(z) and depends analytically on z ∈ Uδ(z0).

By corollary 4.13, it follows that the vectors vj,k(z) = U(z, z0)vj,k, k ∈

1, . . . ,mj, form a basis of the generalized eigenspace Im(P (z)) = ker(λj(z)I−

A(z))mj and that they depend analytically on z ∈ Uδ(z0).

73

4.4 Analytic properties of eigenprojections near critical

points

Following the chapter about perturbed polynomials, the behavior of eigenpro-

jections near critical points will be investigated. Even though the branch points

of the eigenvalues and the eigenprojections are the same, the behavior of the

eigenprojections differ considerably from the behavior of the eigenvalues.

In the following, z0 ∈ Ω denotes a possibly critical point of A(·). λ1, . . . , λl

denote the distinct eigenvalues of A(z0) and m1, . . . ,mj denote the correspond-

ing algebraic multiplicities. Each non-overlapping positively oriented circle

Γj , j ∈ 1, . . . , l, encloses exactly mj eigenvalues of A(z). κj denotes the max-

imal number of distinct eigenvalues, enclosed by each Γj for z ∈ Uoδ (z0). These

eigenvalues are called the λj -group of eigenvalues of A(z) near z0. Furthermore,

the punctured disk Uoδ (z0) ⊆ Ω does not contain any critical point of A(·).

In analogy to the preceding chapter, (λjk1(z), . . . , λjkqjk (z)), k ∈ 1, . . . , hj,

denote for every j in 1, . . . , l the hj different cycles of eigenvalues of constant

multiplicity mjki(z) = mjk(z) = mjk, obtained by analytic continuation in

Uoδ (z0).

Altogether, A(z) has a total ofl∑j=1

κj =l∑j=1

hj∑k=1

qjk distinct eigenvalues for

every z in Uoδ (z0). Taking their multiplicities into account, they add up to a

total number ofl∑j=1

κjhj∑k=1

qjkmjk = n eigenvalues.

Theorem 4.15. Let A : Ω→ Cn×n be an analytic matrix function on a domain

Ω ⊆ C and let A(z0) =l∑j=1

λjPj +Nj be the spectral representation of A(z0) at

some (possibly critical) point z0 ∈ Ω.

Under these assumptions, the following statements hold.

• The eigenprojections and eigennilpotents of A(z) are branches of analytic

functions of z ∈ Ω with at most algebraic singularities at z0 ∈ CA and their

branch points are the same as those of the corresponding eigenvalues.

• On the punctured disk Uoδ (z0) , A(z) can be represented as

A(z) =l∑

j=1

hj∑

k=1

qjk∑

i=1

(λjki(z)Pjki(z)−Njki(z)).

74

• If qjk = 1, the functions λjk1(z), Pjk1(z) and Njk1(z) can be continued

analytically across the whole disk Uδ(z0).

• If qjk ≥ 2, the branches of the eigenprojection cycle

(Pjk1(z), . . . , Pjkqjk (z)) near z0 can be represented by a Laurent-Puiseux

series of the form

Pjkm(z) =∞∑

l=−ljk

Bjkl(ei 2πqjkm

(z − z0)1qjk )l, z ∈ U−δ (z0),

where ljk ∈ N, Bjkl ∈ Cn×n for l ≥ −ljk.

Proof. Step 1: Analyticity of the total projections.

The total projection Pj(z), associated with one of the λj - groups

λjkm(z) : k ∈ 1, . . . , hj,m ∈ 1, . . . , qjk, z ∈ Uoδ ,

is the sum of all the eigenprojections corresponding to the κj eigenvalues of the

λj-group. Due to corollary 4.11, it depends analytically on z in a small disk

with cent re z0.

Step 2: Analyticity of the eigenprojections and eigennilpotents.

By corollary 4.12, the eigenprojection

Pjki(z) :l⊕

j′=1

hj′⊕

k′=1

qj′k′⊕

i′=1

ker(λj′k′i′(z)I −A(z))mj′k′ → ker(λjki(z)I −A(z))mjk

of A(z), corresponding to the eigenvalue branch λjki(z), is defined and analytic

on the (simply connected) cut disk U−δ (z0). The same statement holds for the

associated eigennilpotents

Njkm(z) = (A(z)− λjkm(z)I)Pjkm(z).

In analogy to the eigenvalue branches, the eigenprojection branches Pjki(z) can

be continued analytically across Uoδ (z0). They are linked one with another if

and only if the same is true for the corresponding eigenvalue branches λjki(z)

because of the following reasoning.

Step 2.a: The permutation to which the families of eigenvalues,

eigenprojections and eigennilpotents are subjected, are identical.

Consider the families (λh(z))h∈1,...,c, (Ph(z))h∈1,...,c and (Nh(z))h∈1,...,c

75

consisting ofl∑j=1

κj = c elements. The resolvent R(s, z) of A(z) is defined, in

accordance with formula (8), as

R(s, z) =c∑

h=1

[Ph(z)

(s− λj(z))+mj−1∑

k=1

Nh(z)k

(s− λj(z))k+1

],

where s is assumed to be somewhere distant from the spectrum σ(A(z)) of A(z)

such that s is contained within ρ(A(z)) for z ∈ Uoδ (z0). Let λ1(z), . . . , λp(z)

be a cycle of eigenvalues. The permutation πz0 maps λh(z) to λh+1(z),

h ∈ 1, . . . , p − 1, and λp(z) to λ1(z). However, since the resolvent R(s, z),

which is unique as a meromorphic operator-valued function in s, remains un-

changed by analytic continuation, the permutation must map Ph(z) to Ph+1(z),

h ∈ 1, . . . , p − 1, and Pp(z) to P1(z). The possibility that Ph(z) = Pk(z)

for some h 6= k is excluded by the property Ph(z)Pk(z) = δhkPh(z). Similar

results hold for the eigennilpotents Nh(z), except that some pair in the set

Nh(z) | h ∈ 1, . . . , c may coincide, since they could all be zero.

Therefore, the operatorqjk∑i=1

λjki(z)Pjki(z) associated with the eigenvalue cycle

(λjk1(z), . . . , λjkqjk (z)) is invariant with respect to analytic continuation along

circular arcs in Uoδ (z0). It can be extended analytically across the punctured

disk.

The application

Pj(z) =

hj∑

k=1

qjk∑

i=1

Pjki(z), z ∈ Uoδ (z0)

is the total projection associated with the λj - group, and

A(z) =l∑

j=1

hj∑

k=1

qjk∑

i=1

(λjki(z)Pjki(z)−Njki(z)), z ∈ Uoδ (z0)

is the spectral representation of A(z).

Step 3: Singularities are at most algebraic.

It is evident that

‖Pjki(z)‖ =

∥∥∥∥∥∥∥

1

2πi

∫

Γjki(z)

R(s,A(z))ds

∥∥∥∥∥∥∥≤ r(Γjki(z)) max

s∈Γjki(z)‖R(s,A(z))‖,

where Γjki(z) is a circle around λjki(z) and r(Γjki(z)) is the radius of this circle.

76

Furthermore,

‖R(s,A(z))‖ = ‖(sI −A(z))−1‖ ≤ γ‖sI − A(z)‖n−1

|det(sI −A(z))|≤

≤ γ(‖A(z)‖+ |s|)n−1

∣∣∣∣∣l∏j=1

hj∏k=1

qjk∏i=1

(s− λjki(z))

∣∣∣∣∣

,

where γ > 0 is a constant depending only on the chosen norm. The first

inequality will be proved in step 3.a.

Since λjki(z)z→z0−−−→ λj(z0), k ∈ 1, . . . , hj, i ∈ 1, . . . , qjk, the circles Γjki(z)

must be chosen smaller for smaller |z| in order to ensure that they remain non-

overlapping.

By choosing r(Γjki(z)) = |z|α with an appropriate α > 0, it can be ensured that

∣∣∣∣∣∣

l∏

j=1

hj∏

k=1

qjk∏

i=1

(s− λjki(z))

∣∣∣∣∣∣≥ γ′|z|αn, s ∈ Γjki(z), γ′ > 0

Hence,

‖Pjki(z)‖ ≤ γr(Γjki) maxs∈Γjki

(‖A(z)‖+|s|)n−1∣∣∣∣l∏j=1

hj∏k=1

qjk∏i=1

(s−λjki(z))

∣∣∣∣≤

≤ γ|z|α maxs∈Γjki

(‖A(z)‖+ |s|)n−1

γ′|z|αn≤

≤ β 1|z|(n−1)α ,

where β > 0 is a constant.

Step 3.a: The inequality ‖A−1‖ ≤ γ|det(A)|‖A‖

n−1 holds.

Since all norms in a finite dimensional vector space are equivalent, the following

inequalities are satisfied.

γ′1‖·‖ ≤ ‖·‖max ≤ γ′2‖·‖

γ′′1 ||| · ||| ≤ ||| · |||max ≤ γ′′2 ||| · |||,

where ‖·‖ denotes an arbitrary norm of Kn, ‖x‖max = ‖x1b1 + · · ·+xnbn‖max =

maxj∈1,...,n

|xj | (bj ,∈ 1, . . . , n is an orthonormal basis of Kn) denotes the maxi-

mum norm, ||| · ||| denotes an arbitrary induced operator norm and |||A|||max =

max(j,k)∈1,...,n2

|ajk| denotes the maximum matrix norm, which isn’t an operator

norm.

77

Therefore ,

‖u‖ = ‖A−1v‖

≤1

|det(A)|

∣∣∣∣∣

∣∣∣∣∣

∣∣∣∣∣

[((−1)k+lSkl(A)

)nk,l=1

]T ∣∣∣∣∣

∣∣∣∣∣

∣∣∣∣∣ ‖v‖

≤1

|det(A)|

1

γ′′1max

(j,k)∈1,...,n2|Skl(A)|︸ ︷︷ ︸

=∑

σ∈Πn−1

sign(σ)ai1σ(i1)···ai(n−1)σ(i(n−1))

‖v‖

≤1

|det(A)|

1

γ′′1max

(j,k)∈1,...,n2|ajk|

n−1(n− 1)!‖v‖

≤(n− 1)!

|det(A)|

γ′′2γ′′1|||A|||n−1‖v‖,

where Skl(A) is the determinant of the (n − 1) × (n − 1)-dimensional ma-

trix that results from deleting row k and column j of A. Hence, |||A−1||| ≤

γ|det(A)| |||A|||

n−1, where γ > 0 is a constant.

Theorem 4.16 (Butler). If z0 is a branch point of order (qjk − 1) ≥ 1 of an

eigenvalue cycle (λjk1(z), . . . , λjkqjk (z)) of A(z), it follows that

• the Laurent-Puiseux expansion of the associated eigenprojection Pjki(z) in

powers of (z− z0)1q contains negative powers. Therefore, ‖Pjki(z)‖

z→z0−−−→

∞.

Proof. Pjki(z) belongs to the cycle (Pjk1(z), . . . , Pjkqjk (z)) of eigenprojec-

tions. Analytic continuation along a small circle around z0 changes Pjki(z)

to Pjk(i+1)(z), i ∈ 1, . . . , qjk − 1 and Pjkqjk to Pjk1(z).

If the Laurent Puiseux expansion were not containing negative powers of (z −

z0)1q , it would follow that the limits

limz→z0

Pjk(i+1)(z) = limz→z0

Pjki(z) = Bjk0

exist. But since

• limz→z0

Pjki(z)Pjk(i+1)(z) = 0 = Bjk0Bjk0 and

• limz→z0

Pjki(z)Pjki(z) = Bjk0Bjk0 = limz→z0 Pjki(z) = Bjk0

it follows that Bjk0Bjk0 = Bjk0 = 0.

78

This is a contradiction to the fact that Pjki(z) is a non-zero projection and that

‖Pjki(z)‖ ≥ 1 must hold.

Therefore, analyticity of eigenprojections at branch points cannot be expected.

A property of a matrix family, which ensures that branch points of A(z) does

not exist for z ∈ Ω, is the following one.

Definition 4.17 (Normality condition at a point z0). A continuous matrix

function A : Ω → Cn×n satisfies the normality condition at a point z0 ∈ Ω if

and only if there exists a sequence (zi)i∈N∗ ∈ Ω\z0 which converges towards

z0 and for which A(zi) is normal for every i ∈ N∗.

By continuity it follows that under the normality condition, A(z0) must be

normal as well. The next theorem provides an important analyticity result for

eigenvalues and eigenprojections.

Theorem 4.18 (Rellich). Let A : Ω → Cn×n be an analytic matrix function

which satisfies the normality condition at z0 ∈ Ω. Its spectral representation at

z0 is

A(z0) =l∑

i=1

λjPj .

The eigenvalues λj have algebraic multiplicities mj.Under these conditions, it

follows that

• there are κj ≤ mj analytic eigenvalue functions

λj,k : Uδ(z0)→ C,

which represent the λj - group of eigenvalues of A(z) on Uδ(z0),

λj,1(z), . . . , λj,κj (z), and which coincide on z0 with λj, i.e. λj,k(z0) = λj

for every k ∈ 1, . . . , κj, and that

• there exist κj ≤ mj analytic matrix functions

Pj,k : Uδ(z0)→ Cn×n, k ∈ 1, . . . , κj.

Furthermore, Pj,k(z) is the eigenprojections of A(z) for the eigen-

value λj,k(z), k ∈ 1, . . . , κj,

79

the sum of all eigenprojections Pj,k(z) belonging to the λj-group co-

incides at z = z0 with Pj, i.e.κj∑k=1

Pj,k(z0) = Pj and

each Pj,k(z), j ∈ 1, . . . , l ∧ k ∈ 1, . . . , κj, has constant rank on

Uδ(z0).

• The spectral representation of A(z) for z ∈ Uoδ (z0) is given by

A(z) =l∑

j=1

κj∑

k=1

λj,k(z)Pj,k(z).

Proof. In accordance to the notations introduced at the beginning of section 4.4,

the neighborhood Uδ(z0) ⊆ Ω contains at most one critical point, i.e. Uoδ (z0) ∩

CA = ∅. The non-overlapping circles Γj , j ∈ 1, . . . , l, with cent re λj in C

enclose exactly mj eigenvalues of A(z) for z ∈ Uδ(z0).

Furthermore, (zi)i∈N∗ is a sequence which converges towards z0 and for which

A(zi) is normal, for every i ∈ N∗.

Step 1: The λ(z) are analytic on Uδ(z0)

If z0 is a non-critical point, every λj - group of eigenvalues of A(z) contains only

one, (κj = 1), eigenvalue λj,1(z). This eigenvalue is analytic on the whole disk

Uδ(z0).

If z0 is a critical point, A(zi) is normal and therefore the spectral norm of

all Pj,k(zi), i ∈ N∗ is 1. Since limi→∞‖Pj,k(zi)‖2 = 1, it follows by Butler’s

theorem that z0 isn’t a branch point for any λj,k(z) , i.e. qjk = 1 for every

k ∈ 1, . . . , hj. Therefore, λj,k(z) can be continued analytically across Uδ(z0)

by corollary 3.19.

Step 2: The eigenprojections Pj,k(z) and eigennilpotents Nj,k(z) are

analytic on Uδ(z0).

By corollary 4.12, the eigenprojections Pjk(z) are analytic and they

have constant rank on U−δ (z0). Since z0 isn’t a branch point for

any λj,k(z), the associated eigenprojections Pj,k(z) and eigennilpotent

Njk(z) = (A(z) − λj,k(z)I)Pj,k(z), j ∈ 1, . . . , l, k ∈ 1, . . . , κj can be

extended analytically across Uδ(z0).

80

Step 3: A(z) is diagonalizable.

If A(zi) is normal for all i ∈ N∗, it follows that these matrices are diagonalizable

and hence that all Nj,k(zi) = 0 for all i ∈ N∗. Since z0 is the limit point of the

sequence (zi)i∈N∗ , it follows by the identity theorem that Nj,k(z) is identically

zero on Uδ(z0).

Step 4: The sum of all eigenprojections Pj,k(z) belonging to the

λj-group coincides at z = z0 with Pj .

By continuity, see corollary 4.6, the total projections Pj(z) =κj∑k=1

Pj,k(z) con-

verge to Pj as z converges versus z0, i.e.κj∑k=1

Pj,k(z0) = Pj .

The next corollary is a generalization of Rellich’s local theorem on simply con-

nected subsets.

Corollary 4.19 (Rellich global). Let A : Ω → Cn×n be an analytic matrix

function on Ω and D ⊆ Ω a simply connected subset. A(·) satisfies the normality

condition at each critical point in D.

Under these assumptions, there exist

• a total of N distinct analytic functions λi : D → C and

• N analytic projection-valued functions Pi : D → Cn×n, i ∈ 1, . . . , N,

where N is the number of distinct eigenvalues of A(z) at non-critical points in

Ω.

Furthermore, the spectral representation of A(z) is given for every z ∈ D\CA

by

A(z) =N∑

i=1

λi(z)Pi(z), ∀z ∈ D\CA.

In particular, A(z) is diagonalizable for all z ∈ D.

Proof. By Rellich’s theorem, for every z0 ∈ D the following statements hold for

a small disk Uoδ (z0) ⊆ D.

81

• There are N analytic functions λi : Uδ(z0)→ C such that

σ(A(z)) = λ1(z), . . . , λN (z), ∀z ∈ Uδ(z0).

• There exist N analytic projection-valued functions Pi : Uδ(z0) → Cn×n

such that

for every z ∈ Uoδ (z0), Pi(z) is the eigenprojection of A(z) correspond-

ing to the eigenvalue λi(z), i ∈ 1, . . . , N, and

the spectral representation of A(z) is

A(z) =N∑

i=1

λi(z)Pi(z)

for every z ∈ Uoδ (z0).

Analytic continuation on the whole simply connected subset.

Each of the functions λi(·) and Pi(·) can be continued analytically along any arc

in D in a way that A(z) =N∑i=1

λi(z)Pi(z) is the spectral representation of A(z)

for every non-critical point on the arc. Since D is simply connected this defines,

by the monodromy theorem, N complex functions λi : D −→ C and N analytic

projection-valued functions Pi : D → Cn×n which satisfy the corollary.

An analytic eigenbasis of A(z) can now be constructed with Kato’s global trans-

formation function.

Corollary 4.20. Let A : Ω → Cn×n be an analytic matrix function on Ω and

D ⊆ Ω a simply connected subset. A(·) satisfies the normality condition at each

critical point.

Then, there exists an analytic matrix function

V : D → Cn×n

which satisfies the two following conditions.

• V (z) is invertible, i.e. V (z) ∈ GLn(C)

• V (z)−1A(z)V (z) is diagonal for every z ∈ D

82

Proof. By corollary 4.19, there are N analytic functions λi : D → C, i ∈

1, . . . , N, and N analytic projection valued functions Pj : D → Cn×n, i ∈

1, . . . , N, on D such that A(z) =N∑i=1

λi(z)Pj(z) on D\CA.

By theorem 4.10, for every projection-valued function Pi(z), z ∈ Uδ(zo) exists

an invertible transformation function Ui(z, z′) which depends analytically on

(z, z′) ∈ D ×D.

By corollary 4.13, the vectors Ui(z, z0)vi,l, l ∈ 1, . . . ,mi form a ba-

sis of Im(Pi(z)), whenever (vi,1, . . . , vi,mi) is a basis of Im(Pi(z0)), ∀j ∈

1, . . . , l, ∀k ∈ 1, . . . , κj.

By corollary 4.19, the vectors

Ui(z, z0)vi,l, ∀j ∈ 1, . . . , l, k ∈ 1, . . . , κj, l ∈ 1, . . . ,mi

form an eigenbasis of A(z) on D. Choosing these n vectors as columns of

V (z), it follows that V (z)−1A(z)V (z) is a diagonal matrix, whose entries are

the eigenvalues λi(z) of A(z).

4.5 Real Case

In this last section, the real case will be treated. Therefore, A : I → Cn×n is

analytic on an open interval I ⊆ R and can be expanded into a power series

A(τ) =∞∑k=0

Ak(τ − τ0)k, which is absolutely convergent on a small interval

UR

r(τ0)(τ0).

The complex analytic extension A(z) =∞∑k=0

Ak(z− τ0)k, z ∈ UC

r(τ0)(τ0) does not

contain any non-real critical point, i.e. the number of distinct eigenvalues of

A(τ) at non-real z ∈ UC

r(τ0)(τ0) will be equal to the number of distinct eigenvalues

of A(τ) at non-critical real points τ ∈ I.

The union D of all these disks is a simply connected domain in C with I =

D ∩R. The analytic extension of A(·) to D will be denoted by A(·) as well. By

construction, D does not contain any critical point of A(·) off the real axis.

Corollary 4.21. Let A : I → Cn×n be an analytic matrix function on an open

interval I ⊆ R. The maximum number of distinct eigenvalues of A(·) is N .

If the interval I does not contain any critical points of A(·), then there exists

83

• N analytic functions λj : I → C, j ∈ 1, . . . , N, which represent the

distinct eigenvalues of A(τ) for every τ ∈ I and

• lN analytic projection-valued functions Pj : I → Cn×n, j ∈ 1, . . . , N

representing the corresponding eigenprojections of A(τ) for every τ ∈ I.

If A(·) satisfies the normality condition at all critical points in I, then there

exist again N analytic functions λj : I → C and N analytic projection-valued

functions Pj : I → Cn×n, j ∈ 1, . . . , N, representing the eigenvalues and the

corresponding eigenprojections of A(τ) for every τ ∈ I. Furthermore,

• the spectral representation of A(τ) is given by

A(τ) =N∑

j=1

λj(τ)Pj(τ), τ ∈ I

and

• A(τ) is diagonalizable for every τ ∈ I.

If A(τ) is in addition self-adjoint, i.e. A(τ) = A(τ)∗ for τ ∈ I, then

• the previous statement holds with self-adjoint projections Pj(τ) = Pj(τ)∗ ∈

Kn×n and

• there exists an analytic orthonormal basis of Cn consisting of eigenvectors

of A(·) on I, i.e.

V : I → Cn×n,

where V (τ) is orthogonal and V (τ)−1A(τ)V (τ) is diagonal for every τ ∈ I.

Proof. A(·) is the at the beginning of this section constructed analytic extension

across a simply connected domain D ⊆ C with D ∩ R = I.

Step 1: 1st and 2nd statement

Since the interval I does not contain any critical point, it follows that D does

not contain any critical points. Therefore, the first statement follows from

corollary 4.12 and corollary 3.17 and the second one from corollary 4.19.

84

Step 2: Analytic orthogonal eigenbasis.

Since the eigenprojections of a Hermitian matrix are self-adjoint, it follows that

Pi(τ) = Pi(τ)∗, i ∈ 1, . . . , N, for all non-critical points τ ∈ I and hence by

continuity that the equation is true for all τ ∈ I. Furthermore, the eigenvectors

of self-adjoint matrices are mutually orthogonal.

By the last part of theorem 4.10, it follows that an orthogonal matrix Ui(τ, τ ′),

i ∈ 1, . . . , N, which depends analytically on (τ, τ ′) ∈ I × I exists.

By corollary 4.13, the vectors vik(τ) = Ui(τ, τ ′)vik, k ∈ 1, . . . ,mi form an

orthonormal basis of Im(Pi(τ)), whenever (vi1, . . . , vimi) is an orthonormal basis

of Im(Pi(τ0)),i ∈ 1, . . . , N. This is because Ui(τ, τ ′) is unitary.

Therefore, the vectors vik(τ), i ∈ 1, . . . , N and k ∈ 1, . . . ,mi, form an

orthonormal basis of eigenvectors of A(τ) for all τ ∈ I.

85

5 Bibliography

[1] H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators,

Birkhäuser, 1985.

[2] D. R. Brillinger, Time Series: Data Analysis and Theory, Holden-Day,

San Francisco, 1981.

[3] J. B. Conway, Functions of one complex variable I, Springer, New York,

1978.

[4] M. Deistler and C. Zinner, Modelling High-Dimensional Time Series by

Generalized Linear Dynamic Factor Models: An Introductory Survey, Com-

munications in Information and Systems Vol.7, No.2, pp. 153-166, 2007.

[5] B.A. Fuks, Theory of analytic functions of several complex variables, Amer-

ican Mathematical Soc., Providence, 1965.

[6] I.M. Gelfand, D. A. Raikov, G. E. Shilov, Kommutative normierte

Algebren, VEB Deutscher Verlag der Wissenschaften, 1964.

[7] Hamann E., Deistler M., Scherrer W., Factor Models for Multivariate

Time Series, Adaptive information systems and modelling in economics and

management science pp. 243 - 251, Wien, 2005.

[8] E. J. Hannan and M. Deistler, The Statistical Theory of Linear Sys-

tems, Wiley, New York, 1988.

[9] H. Havlicek, Lineare Algebra für technische Mathematiker, Heldermann,

Lemgo, 2006.

[10] D. Hinrichsen, A. J. Pritchard, Mathematical Systems Theory I - Mod-

elling, State Space Analysis, Stability and Robustness, Springer, Berlin, 2005.

[11] T. Kato, Perturbation theory for linear operators, Springer, Berlin, 1995.

[12] Y. A. Rozanov, Stationary Random Processes, Holden-Day, San Fran-

cisco, 1967.

[13] W. Rudin, Functional analysis, McGraw-Hill, New York, 1998.

86

[14] W. Scherrer and M. Deistler, A structure theory for linear dynamic

errors-in-variables models, SIAM, J. on Control Optimisation, vol. 36(6),

pp. 2148–2175, 1998.

87