Today § 5.1 : Eigenvalues

of § 5.2 : Characteristic Polynomial

Next : Review for Midterm 2

Reminders

My Math Lab Homework # 6 : Due Tomorrow by ltispm

Midterm # 2 : This Wednesday 8- 10pmRom / seat assign .

& practicemidterms posted on course webpage .

Given an nxn matrix A,

an eigenvector is a nonzero

vector v. e- R" with the property that

A v. = tvfor Some scalar XEIR

,called the eigen value

.

• The set of eigenvectors of A for a given eigen value X←

Without s .

B equal to Nul ( ATI ) .

So it B a subspace

of Rn,

called the eigen space for X.

The

eigenvectors are the nonzero vectors in the eigenspahe .

• Typically hand to find the eigenvalues of a matrix ;

once known,

finding the eigenspab is routine.

E 9.Let

A=|§y 6g ) .

2 isaneigenvaluefrlt ;

find abasB for the eigenspad .

Tkeymspalefoz is Null A . ZI ) ←dim ⇒

restitute .tk#Ef 't :#→

k¥31www.t#.sffxM*fxtYtxskB.rHit.HDt 17

KIM?ftEµMl ban .

Eg.

As Hl )

tnde9moahes.pj.YgyAfiIftoHIeHiftssl-fyty.ydeta.oznetwrtble@yjdetthII.o

.

A- It fold , Null :D :{ wyd }espan{ tell dm=t

E.g .

�1� = Idf ) B .v=I± = t foany ten.

↳

Kouyate,

and

its eigenspevis 1122.

ldm = 2)bbasis { ldfl :B

Theorem. Eigenvectors with distinct eigenvalues are

linearly independent .

( Case of zdstnetegenuames

xp Atm )

v. w.

to

Asian ttwgnw Assume { x. d) are Independent.

b W. = Cy for some

A±=µw↳i¥YIa⇒.!

*

So µw.=X± i.e. ( µ. 7) a =o

¥ ¥- Impossible .

Corollary If A is an nxn matrix with n distinct

eigenvahes ,

then there is a basis of Rn

consisting of eigenvectors of A.

D= { s,

I,

.. ; In } basis of e. vectors forA. c)

×' k t Se every meter t.tk

"

can beexpanded as a lm .

Gmb.

I B.

[ Axles ftp.YD⇐ * . Hagey

y =Y,

gtx.at -- - txn.tn

.

=/"

oh, ?yn|µ§n| Ad = x,

As + KAI t - - txnb.vn= X

,X

,9 + ×2X↳ht - .

. tktn.vn

ask.fi#.=fI*d=fgEnleidWhat does that really mean ? D-

II. . ,b eigenvectors

P=[gb . .. b.| l invertible)

AP - AH , g. " b) =[ Ay Ahn . A± )

Ap =p ,=lhbX¥ ... and

i. AMPDM = 14.4 - b) ("

ftp./=PDA2=(PDPY(PDpD=PDP@DPt=PD2p"

Definition :nxn matrices A and B are called similar

IF there B an invertible nxw matrix P with the property

A = PBP-

1 B=PtA@ Q =P- '

i. Q- ' =P↳

.

: AP - PB → i. FHP ' B\B = a apt

• we just saw that, if A has all distinct eigen values

,

then it B similar to a diagonal matrix.

That's important ;

more on that next time.

• Similarity is net the same as now equivalent .

Es . A =/ 11,1 B =/ d,o| row equivalent,

but net similar.

Theorem If A and B are similar,

then they have the

← same eigenvalues ; and the eigenspaces for A haveD= PAF '

ptBp=Athe same dimensions as the eigenspaces for B

.

Let x be an egenoalne of A.

Then the eyen space

for A ± X B Nul ( A- XI ).

I .to .↳FFBEII

's

i. BP , = PHD =xP±

I - Py Is an eigenvector of B y e. value 7.