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Banach J. Math. Anal. (2020) 14:1773–1792https://doi.org/10.1007/s43037-020-00082-x

ORIGINAL PAPER

On the numerical ranges of matrices in max algebra

D. Thaghizadeh1 · M. Zahraei1 · A. Peperko2,3 · N. Haj Aboutalebi4

Received: 6 January 2020 / Accepted: 30 June 2020 / Published online: 18 July 2020 © The Author(s) 2020

AbstractLet M

n(ℝ+) be the set of all n × n nonnegative matrices. Recently, in Tavakolipour

and Shakeri (Linear Multilinear Algebra 67, 2019, https://doi.org/10.1080/03081087.2018.1478946), the concept of the numerical range in tropical algebra was introduced and an explicit formula describing it was obtained. We study the iso-morphic notion of the numerical range of nonnegative matrices in max algebra and give a short proof of the known formula. Moreover, we study several generaliza-tions of the numerical range in max algebra. Let 1 ≤ k ≤ n be a positive integer and C ∈ M

n(ℝ+). We introduce the notions of max k−numerical range and max C−

numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range W

max(�) of a bounded set � of n × n nonnegative matri-

ces is introduced and some of its properties are also investigated.

Keywords Numerical range · Tropical algebra · Max algebra · k-numerical range · C-numerical range

Mathematics Subject Classification 15A60 · 15A18 · 15A80

TusiMathematicalResearchGroup

Communicated by Masatoshi Fujii.

* M. Zahraei [email protected]

D. Thaghizadeh [email protected]

A. Peperko [email protected]; [email protected]

N. Haj Aboutalebi [email protected]

1 Department of Mathematics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran2 Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, 1000 Ljubljana,

Slovenia3 Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia4 Department of Mathematics, Shahrood Branch, Islamic Azad University, Shahrood, Iran

http://crossmark.crossref.org/dialog/?doi=10.1007/s43037-020-00082-x&domain=pdf

1774 D. Thaghizadeh et al.

1 Introduction and preliminaries

The algebraic system max algebra and its isomorphic versions (max plus alge-bra, tropical algebra) provide an attractive way of describing a class of nonlinear problems appearing for instance in manufacturing and transportation scheduling, information technology, discrete event dynamic systems, combinatorial optimiza-tion, mathematical physics, DNA analysis, ...(see e.g. [1, 3, 5, 6, 10, 11] and the references cited there). It has been used to describe these conventionally nonlin-ear problems in a linear fashion.

Max algebra consists of the set of nonnegative real numbers equipped with the basic operations of multiplications a⊗ b = ab, and maximization a⊕ b = max{a, b}. For A = (aij) ∈ Mm×n(ℝ), we say that A is positive (nonnega-tive) and write A > 0 (A ≥ 0) if aij > 0 (aij ≥ 0) for 1 ≤ i ≤ m , 1 ≤ j ≤ n. Let ℝ+ be the set of all nonnegative real numbers and Mm×n(ℝ+) be the set of all m × n non-negative real matrices. The notions Mn(ℝ+) and ℝn

+ are considered for Mn×n(ℝ+)

and Mn×1(ℝ+), respectively. Let A = (aij) ∈ Mm×n(ℝ+) and B = (bij) ∈ Mn×l(ℝ+). The product of A and B in max algebra is denoted by A⊗ B, where (A⊗ B)ij = max

k=1,…,naikbkj. Similarly the vector A⊗ x is defined by

(A⊗ x)i = maxk=1,…,n

aikxk for i = 1,… ,m if x ∈ ℝn+ . If A,B ∈ Mn(ℝ+) then the max

sum A⊕ B in max algebra is defined by (A⊕ B)ij = max{aij, bij} for i, j = 1,… , n . The notation A2

⊗ means A⊗ A, and Ak

⊗ denotes the kth power of A in max algebra.

For A ∈ Mn(ℝ+) and x ∈ ℝn+, let us denote ‖A‖ = max

i,j=1,…,naij and ‖x‖ = max

i=1,…,nxi .

Let rx(A) denote the local spectral radius of A, i.e., rx(A) = lim supj→∞

‖Aj

⊗⊗ x‖1∕j . It

was shown in [11] for x = [x1,… , xn]t ∈ ℝ

n+, x ≠ 0 that rx(A) = lim

j→∞‖Aj

⊗⊗ x‖1∕j

and that rx(A) = max{rei (A) ∶ i = 1,… , n, xi ≠ 0} , where ei denotes the ith stand-ard basis vector and xi denotes the ith coordinate of x. We say that � ≥ 0 is a geo-metric max eigenvalue of A if A⊗ x = 𝜇x for some x ≠ 0 , x ≥ 0 . Let �max(A) denote the set of geometric max eigenvalues of A. The following result of Guna-wardena was restated and reproved in [11, Theorem 2.7].

Theorem 1 If A ∈ Mn(ℝ+) , then

We define the standard vector multiplicity of geometric max eigenvalue � as the number of indices j such that � = rej (A).

The role of the spectral radius of A in max algebra is played by the maximum cycle geometric mean �(A) , which is defined by

and is equal to

�max(A) = {� ∶ there exists j ∈ {1,… , n},� = rej (A)}.

(1)�(A) = max{(ai1ik ⋯ ai3i2ai2i1 )

1∕k ∶ k ∈ ℕ and i1,… , ik ∈ {1,… , n}}

1775On the numerical ranges of matrices in max algebra

A digraph G(A) = (N(A),E(A)) associated to A is defined by setting N(A) = {1,… , n} and letting (i, j) ∈ E(A) whenever aij > 0 . When this digraph contains at least one cycle, one distinguishes critical cycles, where the maximum in (1) is attained. A graph with just one node and no edges will be called trivial. A bit unusually, but in consistency with [3] and [11], a matrix A ∈ Mn(ℝ+) is called irreducible if G(A) is trivial (A is 1 × 1 zero matrix) or strongly connected (for each i, j ∈ N(A) , i ≠ j , there is a path in G(A) that starts in i and ends in j).

It is known that �(A) is the largest geometric max eigenvalue of A, i.e., �(A) = max{� ∶ � ∈ �max(A)} and so we have �(A) = max

j=1,…,nrej (A). Moreover, if A

is irreducible, then �(A) is the unique max eigenvalue and every max eigenvector is positive (see e.g. [3]).

The max permanent of A is

where Sn is the group of permutations on {1,… , n} . The characteristic maxpolyno-mial of A (see e.g. [3, 14, 15]) is a max polynomial

where I denotes the identity matrix. We call its tropical roots (the points of nondiffer-entiability of �A(x) considered as a function on [0,∞) ) algebraic max eigenvalues (or also tropical eigenvalues) of A. The set of all algebraic max eigenvalues is denoted by �trop(A) . For � ∈ �trop(A) its multiplicity as a tropical root of �A(x) (see e.g [3, 14, 15]) is called an algebraic multiplicity of � . It is known that 𝜎max(A) ⊂ 𝜎trop(A)[15, Remark 2.3] and that �(A) = max{� ∶ � ∈ �trop(A)} , but in general, the sets �max(A) and �trop(A) may differ.

Let A ∈ Mn(ℝ+) . The max-numerical range Wmax(A) of A was defined in [15] (actually its isomorphic version in the setting of max-plus (tropical) algebra) and it was shown there that 𝜎trop(A) ⊂ Wmax(A)[15, Theorem 3.10]. It was proved in [15, Theorem 3.7] that given A ∈ Mn(ℝ+)

In the current article we provide a short proof of this fact. This proof provides also new insights, which enables us to consider several generalizations of the max-numerical range and to provide interesting results for these generalizations.

As it will be evident from below the article is partly expository and is organ-ized as follows. In Sect. 2 we give a short proof of the formula (2) (Theorem 2) and obtain some interesting results. In the third section we recall the definition of the joint numerical range of a k-tuple (A1,… ,Ak), where Ai ∈ Mn, i = 1,… , k, and we apply Theorem 2 to obtain a new formula for max joint numerical range Wmax(�) of a bounded set � of n × n nonnegative matrices (11). We move on in Section 4

�(A) = max{(ai1ik ⋯ ai3i2ai2i1 )

1∕k ∶ k ≤ n and i1,… , ik ∈ {1,… , n} mutually distinct}.

perm(A) = max�∈Sn

a1�(1) ⋯ an�(n),

𝜒A(x) = perm(xI ⊕ A),

(2)Wmax(A) =

[min

i∈{1,…,n}aii, max

i,j∈{1,…,n}aij

].


to introduce some definitions and facts, which we need in our proofs and study the max k−numerical range Wk

max(A) , where k ≤ n is a positive integer. We explicitly

describe a formula for Wkmax

(A) (Theorem 3) and then use this to state some of its basic properties (Theorem 4). Related interesting results are also obtained for the max k-geometric spectrum and k-tropical spectrum of A ∈ Mn(ℝ+) . In the last sec-tion we introduce and study the max c−numerical range and max C−numerical range of nonnegative matrices, where c ∈ ℝ

n+ and C ∈ Mn(ℝ+). Also, we investigate some

basic algebraic and geometrical properties of these sets.

2 Max‑numerical range

Let Mn(ℂ) be the vector space of all n × n complex matrices. The numerical range of a square matrix A ∈ Mn(ℂ) is defined by

It is known that W(A) is compact, convex and contains the spectrum of A. In [15], the numerical range of a given square matrix was introduced and described in the setting of max-plus algebra. We study here its isomorphic version in max algebra setting and provide a short proof of one of their main results [15, Theorem 3.7] in Theorem 2.

Definition 1 Let A ∈ Mn(ℝ+) be a non-negative matrix. The max numerical range Wmax(A) of A is defined by

It’s obvious that

Remark 1 (i) If x ∈ ℝn+ , then xt ⊗ x = 1 means

i.e., for all 1 ≤ i ≤ n, 0 ≤ xi ≤ 1 and xj = 1 for some 1 ≤ j ≤ n.

(ii) Suppose that A ∈ Mn(ℝ+) and fA ∶ S ⟶ ℝ+, where

W(A) = {x∗Ax ∶ x ∈ ℂn, x∗x = 1 }

Wmax(A) ={xt ⊗ A⊗ x ∶ x ∈ ℝ

n+, xt ⊗ x = 1

}

Wmax(A) =�xt ⊗ A⊗ x ∶ x ∈ ℝ

n+, xt ⊗ x = 1

�

=

�xt√xt ⊗ x

⊗ A⊗x√

xt ⊗ x∶ 0 ≠ x ∈ ℝ

n+

�

=

�1

xt ⊗ xxt ⊗ A⊗ x ∶ 0 ≠ x ∈ ℝ

n+

�

max{x1, x2,… , xn

}= 1,

S = {x ∈ ℝn+, xt ⊗ x = 1}, fA(x) ∶= xt ⊗ A⊗ x.


So Wmax(A) is the image of the continuous function fA. Since S is a connected set, also Wmax(A) is a connected set.

Next we provide a new short proof of (2) ([15, Theorem 3.7]).

Theorem 2 Let A = (aij) ∈ Mn(ℝ+) be a nonnegative matrix. Then

where a = min1≤i≤n

aii and b = max1≤i,j≤n

aij.

Proof By definition of Wmax(A) we have

Let z ∈ Wmax(A) be given. So,

for some x = [x1, x2,… , xn]t ∈ ℝ

n+ with max{x1, x2,… , xn} = 1. It holds that

0 ≤ xi ≤ 1 for all 1 ≤ i ≤ n and xi0 = 1 for some 1 ≤ i0 ≤ n. By taking

for some 1 ≤ i1, j1 ≤ n, we have

since a = min1≤i≤n

aii and b = max1≤i,j≤n

aij. So Wmax(A) ⊆ [a, b].

To prove the reverse inclusion, recall that the function x ⟼ fA(x) = xt ⊗ A⊗ x is continuous on the compact connected set (Fig. 1)

Let 1 ≤ k ≤ n be such that a = min1≤i≤n

aii = akk . Then the vectors x = [1, 1,… , 1]t and y = ek = [0,… , 0, 1, 0,… , 0]t satisfy x, y ∈ S and

Wmax(A) = [a, b] ⊆ ℝ+,

Wmax(A) = {

n⨁i,j=1

aijxixj ∶ xi ∈ ℝ+ ∀i = 1, 2,… , n , max{x1, x2,… , xn} = 1}.

z = xt ⊗ A⊗ x =

n⨁i,j=1

aijxixj,

z = max{aijxixj ∶ 1 ≤ i, j ≤ n, 0 ≤ xi, xj ≤ 1} = ai1j1xi1xj1 ,

a ≤ ai0i0 = ai0i0xi0xi0 ≤ z = ai1j1xi1xj1 ≤ ai1j1 ≤ b,

(3)S = {x ∈ ℝn+, xt ⊗ x = 1}.

Fig. 1 The max numerical range of a n × n matrix A, Wmax

(A)


and

By Remark 1(ii), Wmax(A) is a connected set and so Wmax(A) = [a, b], which com-pletes the proof. ◻

Remark 2 Alternatively, in Theorem 2, one can prove [a, b] ⊆ Wmax(A), where

in the following constructive way.Let z ∈ [a, b] be given and let a = min

1≤i≤naii = akk and b = max

1≤i,j≤naij = ars for some

1 ≤ k, r, s ≤ n. Now we consider four cases.

Case 1: If r ≠ s, r = k or s = k , by taking x = [z

b,… , 1,… ,

z

b]t, where

xk = 1, xi =z

b∀i ≠ k, we have

Case 2: If r = s = k , then by taking x = [1,… , 1]t we have

Case 3: If r = s , r ≠ k , then by letting � = max{akr, ark} and taking

we have

Case 4: Finally, for r ≠ s , r ≠ k , s ≠ k , by letting � = max{akr, ark, arr} and by taking

if a ≤ z ≤ � , and by taking

fA(x) = xt ⊗ A⊗ x =

n⨁i,j=1

aij = b

fA(y) = yt ⊗ A⊗ y = akk = min1≤i≤n

aii = a.

a = min1≤i≤n

aii, b = max1≤i,j≤n

aij,

xt ⊗ x = 1, xt ⊗ A⊗ x = z.

xt ⊗ x = 1, xt ⊗ A⊗ x = z = a = b.

x =

⎧⎪⎨⎪⎩

[0,… , 1,… ,�

z

b,… , 0]t, xk = 1, xr =

�z

b, xi = 0 ∀i ≠ r, k z >

𝛼2

b

[0,… , 1,… ,z

𝛼,… , 0]t, xk = 1, xr =

z

𝛼, xi = 0 ∀i ≠ r, k z ≤

𝛼2

b,

xt ⊗ x = 1, xt ⊗ A⊗ x = z.

x =

⎧⎪⎨⎪⎩

[0,… , 1,… ,z

𝛼,… , 0]t, xk = 1, xr =

z

𝛼𝛼 = akr ⊕ ark

[0,… , 1,… ,�

z

𝛼,… , 0]t, xk = 1, xr =

�z

𝛼𝛼 = arr, z >

(akr⊕ark)2

𝛼

[0,… , 1,… ,z

akr⊕ark,… , 0]t, xk = 1, xr =

z

akr⊕ark𝛼 = arr, z ≤

(akr⊕ark)2

𝛼


if � ≤ z ≤ b , one can verify that

which completes the proof.

Corollary 1 Let A = diag(�1,… , �n) ∈ Mn(ℝ+) be a diagonal matrix and 0 ≤ �1 ≤ ⋯ ≤ �n, then Wmax(A) = [�1, �n].

Since the maximum of differentiable functions is locally Lipschitz continuous, the following proposition follows.

Proposition 1 Let A = (aij) ∈ Mn(ℝ+) be a nonnegative matrix. The map fA ∶ S ⟶ ℝ+ where

is locally Lipschitz continuous on S.

In conventional algebra, a matrix U ∈ Mn is called unitary if U∗U = UU∗ = In. By analogy one can make the following definition in max algebra:

Definition 2 Let U ∈ Mn(ℝ+) . If Ut⊗ U = U ⊗ Ut = In, then U is called unitary in

max algebra and we denote

The following result was established in [3].

Proposition 2 Let A ∈ Mn(ℝ+) be a non-negative matrix. Then A is unitary in max algebra if and only if A is a permutation matrix.

The following proposition is an analogue of the property of unitary similarity invariance for the field of values, [8, Chapter1]. Its proof is straightforward and it is omitted.

Proposition 3 Let A,P ∈ Mn(ℝ+) be nonnegative matrices and let P be a permuta-tion matrix. Then

For X, Y ⊆ ℝ+, recall that X ⊕ Y is defined as follows:

x = [0,… , 1,… , 1,… ,z

b,… , 0]t, xk = xr = 1, xs =

z

b

xt ⊗ x = 1, xt ⊗ A⊗ x = z,

S = {x ∈ ℝn+, xt ⊗ x = 1}, fA(x) = xt ⊗ A⊗ x

(4)Un = {U ∈ Mn(ℝ+) ∶ Ut⊗ U = U ⊗ Ut = In}.

Wmax(Pt⊗ A⊗ P) = Wmax(A).

X ⊕ Y = {x⊕ y ∶ x ∈ X, y ∈ Y}.


In the following two results we collect some properties of Wmax. By Theorem 2, the proofs are straightforward and we omit them. Let us point out that (ii), (iv) and one inclusion in (i) from Proposition 4 have already been stated in [15].

Proposition 4 Let A,B ∈ Mn(ℝ+) be nonnegative matrices and let �, � ∈ ℝ+ . Then the following statements hold.

(i) Wmax(A⊕ B) = Wmax(A)⊕Wmax(B).

(ii) I f � ≠ 0, Wmax(𝛼A⊕ 𝛽I) = 𝛼Wmax(A)⊕ {𝛽} = 𝛼Wmax(A⊕𝛽

𝛼I). A l s o ,

Wmax(�I) = {�}.

(iii) If A =

[D 0n1×n2

0n2×n1 C

], where D ∈ Mn1

(ℝ+),C ∈ Mn2(ℝ+), n = n1 + n2, then

The equality holds, when min1≤i≤n1

dii = min1≤i≤n2

cii.

(iv) 𝜎max(A) ⊆ 𝜎trop (A) ⊆ Wmax(A).

(v) Wmax(At) = Wmax(A).

(vi) If max1≤i,j≤n

aij = akk for some 1 ≤ k ≤ n , then max(Wmax(A

m⊗)

)= am

kk.

Proposition 5 Let A,B ∈ Mn(ℝ+) be diagonal matrices. If Wmax(A) ⊆ Wmax(B), then

It turns out that in several cases the quotients l(Wmax(Am+1⊗

))

l(Wmax(Am⊗))

and l(Wmax(Am+2⊗

))

l(Wmax(Am⊗))

, where l(⋅) denotes the length of an interval, have interesting asymptotic behaviour. We illus-trate this in the following examples.

Example 1 Let A =

⎡⎢⎢⎣

4 2 5

3 7 1

9 6 8

⎤⎥⎥⎦. So Wmax(A) = [4, 32] and by computing

A2⊗,A3

⊗,A4

⊗,… ,A15

⊗ , respectively, we have

and

By a straightforward induction, we can show that

and so it follows that

Wmax(D)⊕Wmax(C) ⊆ Wmax(A).

Wmax(Am⊗) ⊆ Wmax(B

m⊗), ∀ m ≥ 1.

Wmax(A2⊗) = [32 × 5, 32 × 23], Wmax(A

3⊗) = [73, 32 × 26], Wmax(A

4⊗) = [74, 32 × 29],… ,

Wmax(A15⊗) = [11.25 × 813, 72 × 813].

Am⊗=

⎡⎢⎢⎣

45 × 8m−2 30 × 8m−2 40 × 8m−2

1080 × 8m−4 11.25 × 8m−2 15 × 8m−2

72 × 8m−2 48 × 8m−2 64 × 8m−2

⎤⎥⎥⎦∀m ∈ ℕ,m ≥ 15


It follows from (5) that

Example 2 Let A =

⎡⎢⎢⎢⎣

3 4 2 9

6 2 7 3

9 3 4 5

8 2 5 6

⎤⎥⎥⎥⎦. By induction, we establish that

where

A straightforward computation shows that (6) holds for m = 3 . Assume now that (6) holds for m = 3,… , k . Thus, by the inductive assumption, we obtain

and similarly

Using (6) we have

and

It follows

and

(5)Wmax(Am⊗) = [11.25 × 8m−2, 72 × 8m−2] ∀m ∈ ℕ,m ≥ 15.

limm→∞

l(Wmax(Am+1⊗

))

l(Wmax(Am⊗))

= 8.

(6)A2m⊗

= 72m−2A4⊗, A2m+1

⊗= 72m−2A5

⊗, ∀ m ∈ ℕ, m ≥ 3,

A4⊗=

⎡⎢⎢⎢⎣

5184 1728 3240 3888

4536 1728 2835 3888

3888 2592 2430 5832

3456 2304 2160 5184

⎤⎥⎥⎥⎦, A5

⊗=

⎡⎢⎢⎢⎣

31104 20736 19440 46656

31104 18144 19440 40824

46656 15552 29160 34992

41472 13824 25920 31104

⎤⎥⎥⎥⎦.

A2(k+1)

⊗=A2k

⊗⊗ A2

⊗= 72k−2A4

⊗⊗ A2

⊗

=72k−2A6⊗= 72k−2 × 72A4

⊗

=72k−1A4⊗,

A2(k+1)+1

⊗= 72k−1A5

⊗.

Wmax(A2m⊗) = [72m−2 × 1728, 72m−2 × 5832]

Wmax(A2m+1⊗

) = [72m−2 × 18144, 72m−2 × 46656].

limm→∞

l(Wmax(A2m+1⊗

))

l(Wmax(A2m⊗))

=132

19, lim

m→∞

l(Wmax(A2m+2⊗

))

l(Wmax(A2m+1⊗

))=

114

11


So the limit limm→∞

l(Wmax(Am+1⊗

))

l(Wmax(Am⊗))

does not exist, but the limit

exists and it is equal to 72.

The Cyclicity theorem in max-algebra ([3, Theorem 8.3.5]) states the follow-ing: if A ∈ Mn(ℝ+) is an irreducible matrix, then there exists p ∈ ℕ and there exists T ∈ ℕ such that

holds for every m ≥ T . A matrix A for which there exist p and T such that (7) holds for all m ≥ T is called ultimately periodic. Thus every irreducible matrix is ulti-mately periodic. The smallest p such that (7) holds for all m ≥ T and some T is called a period of A. It is known that a period of an irreducible matrix A equals the cyclicity of A (see[3, Chapter 8]).

More generally, the General cyclicity theorem ([3, Theorem 8.6.9]) states that A ∈ Mn(ℝ+) is ultimately periodic if and only if each irreducible diagonal block of the Frobenius normal form of A has the same geometric max eigenvalue (equal to �(A) ). For definitions, we refer to[3].

Consequently, if A ∈ Mn(ℝ+) is irreducible, then there exist natural numbers p and T such that l(Wmax(A

m+p

⊗)) = 𝜇(A)pl(Wmax(A

m⊗)) for all m ≥ T . Therefore, if

l(Wmax(Am⊗)) > 0 for all m ≥ T , then

In the following, we consider two special cases when (8) holds for p = 1.

Proposition 6 Let A = (aij) ∈ Mn(ℝ+) be a nonnegative matrix such that l(Wmax(A)) > 0 .

(i) If A is an upper triangular matrix or a lower triangular matrix, then

limm→∞

l(Wmax(A2m+2⊗

))

l(Wmax(A2m⊗))

= limm→∞

l(Wmax(A2m+3⊗

))

l(Wmax(A2m+1⊗

))= 72.

limm→∞

l(Wmax(Am+2⊗

))

l(Wmax(Am⊗))

(7)Am+p

⊗= 𝜇(A)pAm

⊗

(8)limm→∞

l(Wmax(Am+p

⊗))

l(Wmax(Am⊗))

= 𝜇(A)p.

limm→∞

l(Wmax(Am+1⊗

))

l(Wmax(Am⊗))

= max1≤i≤n

aii;


(ii) If max1≤i,j≤n

aij = max1≤i≤n

aii and the limit limm→∞

l(Wmax(Am+1⊗

))

l(Wmax(Am⊗))

exists, then it is equal

to the maximum of aii on 1 ≤ i ≤ n.

Proof (i) By Propositions 3 and 4(v) we may assume without loss of generality that a11 ≤ a22 ≤ … ≤ ann and that A is upper triangular matrix. By computing Am

⊗ , one

can see that there exists some s ≥ n such that

We claim that there exist 1 ≤ i0 ≤ n and s0 ≥ n such that

If this is not the case, then for some large enough m ≥ n, there exist 1 ≤ i1, j1 ≤ n such that

This shows that

which leads to a contradiction. This shows that the claim is true and the result fol-lows since the minimal element on the diagonal of Am

⊗ is strictly smaller than am

nn for

all m ≥ s0.(ii) We may assume that a11 ≤ a22 ≤ ⋯ ≤ ann . As max

1≤i,j≤naij = max

1≤i≤naii = ann, it

follows that

It now follows from (9) that

which completes the proof. ◻

Am⊗=

⎡⎢⎢⎢⎢⎢⎢⎣

am11

k12am−122

⋯ k1iam−i+1ii

⋯ k1nam−n+1nn

0 am22

⋯ k2iam−i+2ii

⋯ k2nam−n+2nn

⋮ 0 ⋱ ⋮ ⋯ ⋮

0 ⋮ 0 amii

⋯ kinam−n+inn

⋮ ⋮ ⋯ ⋱ ⋱ ⋮

0 0 ⋯ ⋯ 0 amnn

⎤⎥⎥⎥⎥⎥⎥⎦

∀ m ≥ s.

max1≤i,j≤n

(Am⊗)ij = ki0na

m−n+i0nn ∀ m ≥ s0.

ki1j1am−j1+i1j1j1

> kinam−n+inn

, 1 ≤ i ≤ n.

m <

ln

(kina

j1−i1j1 j1

ki1 j1an−inn

)

ln(

aj1 j1

ann

) ,

(9)amnn

= max1≤i=i0≤i1≤⋯≤im−1≤j=im≤n

aii1ai1i2 … aim−1j, ∀ m ≥ 1.

limm→∞

l(Wmax(Am+1⊗

))

l(Wmax(Am⊗))

= limm→∞

am+1nn

− am+111

amnn− am

11

= ann,


3 Max joint numerical ranges

Recall that the joint numerical range of a k-tuple (A1,… ,Ak), where Ai ∈ Mn, i = 1,… , k, is defined by ([2, 12])

where S1 = {x ∈ ℂn ∶ x∗x = 1}. So, one can define the max joint numerical range of

a k-tuple of n × n nonnegative matrices � = (A1,… ,Ak) in the following way:

Remark 3 Note that it follows from the above definition that (a1,… , ak) ∈ Wmax(A1,… ,Ak) if and only if there exists x ∈ ℝ

n+, xt ⊗ x = 1 such

that ai = xt ⊗ Ai ⊗ x, for all 1 ≤ i ≤ k.

From Theorem 2 and Remark 3 we conclude the following result.

Corollary 2 If � = (A1,… ,Ak) such that Al = (a(l)

ij) ∈ Mn(ℝ+), l = 1,… , k, then

where tl = max1≤i,j≤n

a(l)

ij and sl = min

1≤i≤na(l)

ii for all 1 ≤ l ≤ k. Consequently, Wmax(�) is a

compact set.

Next we define a max joint numerical range Wmax(�) of a bounded set � of n × n nonnegative matrices in the following way:

By Theorem 2, we have

Recall that the supremum matrix S(�) is defined by

and that the generalized (joint) spectral radius �(�) of � is equal to ([5, 6, 9, 10, 13])

W(A1,… ,Ak) = {(x∗A1x,… , x∗Akx) ∶ x ∈ S1},

(10)Wmax(𝔸) = {(xt ⊗ A1 ⊗ x,… , xt ⊗ Ak ⊗ x) ∶ x ∈ ℝn+, xt ⊗ x = 1}.

Wmax(𝔸) ⊆ [s1, t1] ×⋯ × [sk, tk] ⊆ ℝk+,

Wmax(�) = ∪A∈�Wmax(A).

(11)Wmax(�) =⋃A∈�

[min1≤i≤n

aii, max1≤i,j≤n

aij

].

S(�)ij = supA∈�

aij

𝜇(𝛴) = lim supm→∞

(supA∈𝛴m

⊗

𝜇(A)

) 1

m

= supm∈ℕ

(supA∈𝛴m

⊗

𝜇(A)

) 1

m

,


where 𝛴m⊗= {A1 ⊗ A2 ⊗…⊗ Am ∶ Ai ∈ 𝛴} . The max Berger Wang formula

asserts that ([9, 10, 13])

where ‖A‖ = maxi,j=1,…,n aij (since all norms on ℝn×n are equivalent one can use here any norm on ℝn×n ). We also have �(�) = �(S(�)) , ‖�‖ = sup

A∈�,i,j=1,…,n

aij = ‖S(�)‖ and

where tr⊗(A) = max

i=1,…,naii ([10, Theorem 3.3.]).

By (11), the following result follows.

Corollary 3 If � is a bounded set of n × n nonnegative matrices, then

4 k‑numerical range, k‑geometric max spectrum and k‑tropical spectrum

Now, we introduce and study the max k−numerical range, where k ≤ n is a positive integer. Let Ik denote the k × k identity matrix. A matrix X ∈ Mn×k(ℝ+) is called an isometry in max algebra if Xt

⊗ X = Ik, and the set of all n × k isometry matrices in max algebra is denoted by Xn×k. For the case k = n, Xn×n is equal to Un, which was introduced in Definition 2.

Definition 3 For A ∈ Mn(ℝ+) with k ≤ n, the max k−numerical range of A is defined and denoted by

It is clear that W1max

(A) = Wmax(A), so the notion of max k−numerical range is a generalization of the max numerical range of matrices.

Note that

where 1 ≤ i, j ≤ k and xi, xj ∈ ℝn+,

𝜇(𝛴) = limm→∞

�supA∈𝛴m

⊗

‖A‖� 1

m

= infm∈ℕ

�supA∈𝛴m

⊗

‖A‖� 1

m

,

(12)𝜇(𝛴) = lim supm→∞

(supA∈𝛴m

⊗

tr⊗(A)

) 1

m

,

infA∈�,i=1,…,n

aii = infWmax(�) ≤ �(�) ≤ supWmax(�) = ‖�‖

Wkmax

(A) =

{k⨁

i=1

xti⊗ A⊗ xi ∶ X = [x1, x2,… , xk] ∈ Xn×k

}

={tr⊗(Xt

⊗ A⊗ X) ∶ X = [x1, x2,… , xk] ∈ Xn×k

}.

tr⊗(Xt

⊗ A⊗ X) = xt1⊗ A⊗ x1 ⊕ xt

2⊗ A⊗ x2 ⊕⋯⊕ xt

k⊗ A⊗ xk,


Remark 4 For a nonnegative matrix A = (aij) and 1 ≤ k ≤ n , the map fA ∶ Xn×k ⟶ ℝ+ is locally Lipschitz continuous on Xn×k, where

Note that Wkmax

(A) is the image of the continuous function fA. Using connectivity and compactness of Xn×k, W

kmax

(A) is a connected and compact set.

We have the following explicit formula for Wkmax

(A).

Theorem 3 Suppose that A = (aij) ∈ Mn(ℝ+) and let 1 ≤ k ≤ n be a positive integer. We have

where c = min{

k⨁j=1

aijij ∶ 1 ≤ i1 < i2 < ⋯ < ik ≤ n} and d = max1≤i,j≤n

aij.

Proof By Definition 3 and Theorem 2, it follows that minWkmax

(A) = c and maxWk

max(A) = d . Since Wmax is a connected set, by Remark 4, this implies (13).

◻

In the following theorem, we state some basic properties of the max k−numeri-cal range of matrices.

Theorem 4 Let A ∈ Mn(ℝ+) and 1 ≤ k ≤ n . Then the following assertions hold:

(i) Wkmax

(𝛼A⊕ 𝛽I) = 𝛼Wkmax

(A)⊕ 𝛽 and Wkmax

(A⊕ B) ⊆ Wkmax

(A)⊕Wkmax

(B), where �, � ∈ ℝ+ and B ∈ Mn(ℝ+);

(ii) Wkmax

(Ut⊗ A⊗ U) = Wk

max(A) if U ∈ Un;

(iii) f B ∈ Mm(ℝ+) is a principal submatrix of A and k ≤ m, then Wkmax

(B) ⊆ Wkmax

(A). Consequently, if V = [ei1 , ei2 ,… , eis ] ∈ Mn×s(ℝ+), where 1 ≤ s ≤ n , then Wk

max(Vt

⊗ A⊗ V) ⊆ Wkmax

(A), and the equality holds if s = n;(iv) Wk

max(At) = Wk

max(A);

(v) If k < n, then Wk+1max

(A) ⊆ Wkmax

(A). Consequently,

Proof The properties (i), (ii), (iii) and (iv) easily follow from Theorem 3 (or directly from Definition 3).

xti⊗ xj = 𝛿ij =

{1 i = j

0 i ≠ j.

Xn×k = {X ∈ Mn×k(ℝ+) ∶ Xt⊗ X = Ik}, fA(X) ∶= tr

⊗(Xt

⊗ A⊗ X)

(13)Wkmax

(A) = [c, d],

Wnmax

(A) ⊆ Wn−1max

(A) ⊆ ⋯ ⊆ W2max

(A) ⊆ Wmax(A).


To prove (v), let z ∈ Wk+1max

(A) be given. So, by Definition 3, there exist 1 ≤ i1 ≤ i2 ≤ … ≤ ik ≤ ik+1 ≤ n, X = [xi1 , xi2 ,… , xik , xik+1 ] ∈ Xn×(k+1) such that

Now, one can assume that, without loss of generality,

Hence, by setting X = [xi2 ,… , xik , xik+1], we have X ∈ Xn×k and so

This implies that z ∈ Wkmax

(A) and the proof is complete. ◻

The following example illustrates Theorem 4 (v).

Example 3 If

then

Similarly as in the classical linear case, we define below max k-geometric spec-trum and k-tropical spectrum of A ∈ Mn(ℝ+).

Definition 4 Let A ∈ Mn(ℝ+) , 1 ≤ k ≤ n and let �1, ...,�n ∈ �max(A) counting stand-ard vector multiplicities. The max k-geometric spectrum of A is defined by

Definition 5 Let A ∈ Mn(ℝ+) , 1 ≤ k ≤ n and let �1, ..., �n ∈ �trop(A) counting tropi-cal multiplicities. The k-tropical max spectrum of A is defined by

It is clear that �1max

(A) = �max(A) and �1trop

(A) = �trop(A).

z =

ik+1⨁i=i1

xti⊗ A⊗ xi.

xti1⊗ A⊗ xi1 ≤ ⋯ ≤ xt

ik⊗ A⊗ xik ≤ xt

ik+1⊗ A⊗ xik+1 .

z =

ik+1⨁i=i2

xti⊗ A⊗ xi.

A =

⎡⎢⎢⎢⎣

4 7 5 8

8 2 0 7

2 8 1 4

1 6 2 2

⎤⎥⎥⎥⎦,

W1max

(A) = Wmax(A) = [1, 8], W2max

(A) = [2, 8], W3max

(A) = [2, 8], W4max

(A) = [4, 8].

𝜎kmax

(A) =

{k⨁

j=1

𝜇ij∶ 1 ≤ i1 < i2 < ⋯ < ik ≤ n

}.

𝜎ktrop

(A) =

{k⨁

j=1

𝜆ij∶ 1 ≤ i1 < i2 < ⋯ < ik ≤ n

}.


Recall that the max convex hull of a set M ⊆ ℝ+, which is denoted by conv⊗(M),

is defined as the set of all max convex linear combinations of elements of M, i.e.,

By Definitions 4 and 5, it is obvious that

Since 𝜎max(A) ⊂ 𝜎trop(A) ⊂ Wmax(A) , the following result follows from Definitions 4, 5 and 3 and Theorems 2 and 3.

Proposition 7 Let A ∈ Mn(ℝ+) and 1 ≤ k ≤ n . Then conv⊗(𝜎k

max(A)) ⊂ Wk

max(A) and

conv⊗(𝜎k

trop(A)) ⊂ Wk

max(A).

By Theorem 3, (14) and (15), we have the following two results.

Proposition 8 Let A ∈ Mn(ℝ+), 1 ≤ k ≤ n and let �1, ...,�n ∈ �max(A) counting standard vector multiplicities. If min

1≤i1<⋯<ik≤n⊕𝜇ij

= min1≤i1<⋯<ik≤n

⊕aijij and

�(A) = max1≤i,j≤n

aij, then

Proposition 9 Let A ∈ Mn(ℝ+), 1 ≤ k ≤ n and let �1, ..., �n ∈ �trop(A) counting tropi-cal multiplicities. If min

1≤i1<⋯<ik≤n⊕𝜆ij

= min1≤i1<⋯<ik≤n

⊕aijij and �(A) = max1≤i,j≤n

aij , then

Remark 5 Let A ∈ Mn(ℝ+) be irreducible and let 1 ≤ k ≤ n . Then �(A) is a unique geometric max eigenvalue of A, and so for all 1 ≤ k ≤ n, we have

where B = (bij) ∈ Mn(ℝ+) such that bii = �(A) = max1≤i,j≤n

bij, 1 ≤ i ≤ n.

In the following result we state the relationship between the max k−geometric spectrums of A.

Proposition 10 Let A ∈ Mn(ℝ+) and 1 ≤ k < n . Then 𝜎k+1max

(A) ⊆ 𝜎kmax

(A). Consequently,

conv⊗(M) ∶= {

m⨁i=1

𝛼ixi ∶ m ∈ ℕ, xi ∈ M, 𝛼i ≥ 0, i = 1,… ,m,

m⨁i=1

𝛼i = 1}.

(14)conv⊗(𝜎k

max(A)) =

[min

⨁1≤i1<i2<⋯<ik≤n

𝜇ij, max

1≤i≤n𝜇i

], ∀ 1 ≤ k ≤ n, 1 ≤ j ≤ k,

(15)conv⊗(𝜎k

trop(A)) =

[min

⨁1≤i1<i2<⋯<ik≤n

𝜆ij, max

1≤i≤n𝜆i

], ∀ 1 ≤ k ≤ n, 1 ≤ j ≤ k.

conv⊗(𝜎k

max(A)) = Wk

max(A).

conv⊗(𝜎k

trop(A)) = Wk

max(A).

�kmax

(A) = Wkmax

(B) = {�(A)},


Proof At first, let z ∈ �k+1max

(A) be given. By Definition 4 there exists

1 ≤ i1 < i2 < ⋯ < ik+1 ≤ n such that z =

k+1⨁j=1

�ij. Since

z =

k+1⨁j=1

�ij=

k⨁p=1

�jp∈ �

kmax

(A) , the proof is complete. ◻

The following example from [11] illustrates the result above.

Example 4 Let

Then [11, Example 2.13.],

where the rej (A) ’s are the max geometric eigenvalues of A. One can easily check that

Also

conv⊗(𝜎4

max(A)) = {3} and conv

⊗(𝜎5

max(A)) = {3}.

The following analogue of Proposition 10 for the k−tropical spectrum is proved in a similar way as Proposition 10.

Proposition 11 Let A ∈ Mn(ℝ+) and 1 ≤ k < n . Then 𝜎k+1trop (A) ⊆ 𝜎

ktrop

(A). Consequently,

Remark 6 It is also possible to consider the max algebra analogues of a related object from the usual algebra, namely the max algebra analogues of a higher-rank numerical range of a matrix B (cf. [4, 16]). We do not study these objects in this arti-cle and leave this for further research.

{𝜇(A)} = 𝜎nmax

(A) ⊆ 𝜎n−1max

(A) ⊆ ⋯ ⊆ 𝜎2max

(A) ⊆ 𝜎⊗(A).

A =

⎡⎢⎢⎢⎢⎣

2 0 0 0 0

1 3 0 0 0

1 0 1 1 0

0 0 1 2 0

0 0 0 0 1

⎤⎥⎥⎥⎥⎦.

re1 (A) = re2 (A) = 3, re3 (A) = re4 (A) = 2, re5 (A) = 1,

�5max

(A) = {3}, �4max

(A) = {3}, �3max

(A) = {2, 3}, �2max

(A) = {2, 3}, �1max

(A) = {1, 2, 3}.

conv⊗(𝜎1

max(A)) = [1, 3], conv

⊗(𝜎2

max(A)) = [2, 3], conv

⊗(𝜎3

max(A)) = [2, 3],

{𝜇(A)} = 𝜎ntrop

(A) ⊆ 𝜎n−1trop

(A) ⊆ ⋯ ⊆ 𝜎2trop

(A) ⊆ 𝜎trop(A).


5 Max c‑numerical range and max C‑numerical range

Let A,C ∈ Mn(ℝ+) and c ∈ ℝn+. Next we define and study max c-numerical range

and max C-numerical range of A ∈ Mn(ℝ+) . To access more information about some known results in the complex case, see [8, Chapter1].

Definition 6 Let A ∈ Mn(ℝ+) and c = [c1, c2,… , cn]t ∈ ℝ

n+. The max c−numerical

range of A is defined and denoted by

In Definition 6, it’s obvious that X ∈ Un, where the notation Un was denoted in (2).

It is clear that Wcmax

(A) = {tr⊗(C⊗ Xt

⊗ A⊗ X) ∶ X ∈ Un}, where C = diag(c1,… , cn), c = [c1, c2,… , cn]

t ∈ ℝn+. Also, one can easily verify

and, consequently,

Remark 7 Let A ∈ Mn(ℝ+) and c = [�,… , �]t ∈ ℝn+. Then

Since tr⊗(Xt

⊗ A⊗ X) = tr⊗(A) for X ∈ Un it follows that

Next we introduce and study the notion of max C−numerical range of non-negative matrices, where C ∈ Mn(ℝ+).

Definition 7 Let A,C ∈ Mn(ℝ+). The max C−numerical range of A is defined and denoted by

Example 5 Let C = (cij) ∈ Mn(ℝ+) such that c11 = 1 and cij = 0 elsewhere. Then one can easily obtain that

In the following theorem, we state some basic properties of the max C−numer-ical range of non-negative matrices.

Wcmax

(A) = {

n⨁i=1

cixti⊗ A⊗ xi ∶ X = [x1, x2,… , xn] ∈ Mn(ℝ+), X

t⊗ X = In}.

Wcmax

(A) = {ck(⊕ni=1

aii) ∶ k = 1, 2,… , n},

conv⊗(Wc

max(A)) = [min

1≤k≤nck(⊕

ni=1

aii), ⊕nk=1

ck(⊕ni=1

aii)].

Wcmax

(A) = {tr⊗(𝛼Xt

⊗ A⊗ X) ∶ X ∈ Un}.

Wcmax

(A) = {𝛼tr⊗(A)}.

WCmax

(A) = {tr⊗(C⊗ Xt

⊗ A⊗ X) ∶ X = [x1, x2,… , xn] ∈ Un}

conv⊗(WC

max(A)) = [min

1≤i≤naii,⊕

ni=1

aii].


Theorem 5 Let A,C ∈ Mn(ℝ+). Then the following assertions hold:

(i) WCmax

(𝛼A⊕ 𝛽In) = 𝛼WCmax

(A)⊕ 𝛽tr⊗(C), where �, � ∈ ℝ+;

(ii) WCmax

(A⊕ B) ⊆ WCmax

(A)⊕WCmax

(B) and WC⊕Dmax

(A) ⊆ WCmax

(A)⊕WDmax

(A), where B,D ∈ Mn(ℝ+);

(iii) WCmax

(Ut⊗ A⊗ U) = WC

max(A), where U ∈ Un;

(iv) If Ct = C, then WCmax

(At) = WCmax

(A);

(v) If C = �In, where � ∈ ℝ+, then WCmax

(A) = {𝛼tr⊗(A)}.

(vi) WCmax

(A) = WAmax

(C).

(vii) WVt⊗C⊗V

max(A) = WC

max(A), where V ∈ Un.

Proof The assertions (i), (ii), (iii), (iv), (v) and (vi) follow easily from Definition 7.To prove (vii), let z ∈ WVt

⊗C⊗Vmax

(A) be given. Then by Definition 7, there exists X ∈ Un, such that z = tr

⊗(Vt

⊗ C⊗ V ⊗ Xt⊗ A⊗ X). Since

By setting U = X ⊗ Vt ∈ Un , one has z ∈ WCmax

(A), and so

In a similar way the reverse inclusion can easily be verified. ◻

The following result is a special case of Theorem5.

Corollary 4 Let A ∈ Mn(ℝ+) and c = [c1,… , cn]t ∈ ℝ

n+. Then the following asser-

tions hold:

(i) Wcmax

(𝛼A⊕ 𝛽In) = 𝛼Wcmax

(A)⊕ 𝛽

n⨁i=1

ci, where �, � ∈ ℝ+;

(ii) Wcmax

(Ut⊗ A⊗ U) = Wc

max(A), where U ∈ Un;

(iii) Wcmax

(A⊕ B) ⊆ Wcmax

(A)⊕Wcmax

(B) and Wc⊕dmax

(A) ⊆ Wcmax

(A)⊕Wdmax

(A), where d ∈ ℝ

n+ and B ∈ Mn(ℝ+);

(iv) Wcmax

(At) = Wcmax

(A);

(v) If A = �In, where � ∈ ℝ+, then Wcmax

(A) = {�

n⨁i=1

ci}.

Acknowledgements The third author was supported in part by the Slovenian Research Agency (grants P1-0222, J1-8133, J1-8155, and N1-0071). We thank the referees for very useful comments and corrections.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-mons licence, and indicate if changes were made. The images or other third party material in this article

tr⊗

(Vt

⊗ C⊗ V ⊗ Xt⊗ A⊗ X

)= tr

⊗

(C⊗ (X ⊗ Vt)t ⊗ A⊗ (X ⊗ Vt)

).

WVt⊗C⊗V

max(A) ⊆ WC

max(A).


are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.

References

1. Aghamollaei, G.H., Ghasemizadeh, N.: Some results on matrix polynomials in the max algebra. Banach J. Math. Anal. 9, 17–26 (2015)

2. Binding, P., Li, C.K.: Joint ranges of Hermitian matrices and simultaneous diagonalization. Linear Algebra Appl. 151, 157–167 (1991)

3. Butkovič, P.: Max-linear Systems: Theory and Algorithms. Springer-Verlag, London (2010) 4. Gau, H.L., Wu, P.Y.: Higher-rank numerical ranges and Kippenhahn polynomials. Linear Algebra

Appl. 438, 3054–3061 (2013) 5. Guglielmi, N., Mason, O., Wirth, F.: Barabanov norms, Lipschitz continuity and monotonicity for

the max algebraic joint spectral radius. Linear Algebra Appl. 550, 37–58 (2018) 6. Gursoy, B.B., Mason, O.: Spectral properties of matrix polynomials in the max algebra. Linear

Algebra Appl. 435, 1626–1630 (2011) 7. Gutkin, E., Jonckheere, E.A., Karow, M.: Convexity of the joint numerical range: topological and

differential geometric viewpoints. Linear Algebra Appl. 376, 143–171 (2004) 8. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge

(1991) 9. Lur, Y.Y.: A max version of the generalized spectral radius theorem. Linear Algebra Appl. 418,

336–346 (2006) 10. Müller, V., Peperko, A.: Generalized spectral radius and its max algebra version. Linear Algebra

Appl. 439, 1006–1016 (2013) 11. Müller, V., Peperko, A.: On the spectrum in max algebra. Linear Algebra Appl. 485, 250–266

(2015) 12. Müller, V., Tomilov, Y.: Joint numerical ranges and compressions of powers of operators. J. Lond.

Math. Soc. 99, 127–152 (2019) 13. Peperko, A.: On the max version of the generalized spectral radius theorem. Linear Algebra Appl.

428, 2312–2318 (2008) 14. Rosenmann, A., Lehner, F., Peperko, A.: Polynomial convolutions in max-plus algebra. Linear

Algebra Appl. 578, 370–401 (2019) 15. Tavakolipour, H., Shakeri, F.: On the numerical range in tropical algebra. Linear Multilinear Alge-

bra 67, (2019). https ://doi.org/10.1080/03081 087.2018.14789 46 16. Woerderman, H.: The higher rank numerical range is convex. Linear Multilinear Algebra 56, 65–67

(2008)

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