On the numerical ranges of matrices in max algebra

Let Mn(R+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{n}({\mathbb {R}}_{+})$$\end{document} be the set of all n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document} 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 isomorphic notion of the numerical range of nonnegative matrices in max algebra and give a short proof of the known formula. Moreover, we study several generalizations of the numerical range in max algebra. Let 1≤k≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le k \le n$$\end{document} be a positive integer and C∈Mn(R+).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C \in M_{ n}({\mathbb {R}}_{+}).$$\end{document} We introduce the notions of max k-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k-$$\end{document}numerical range and max C-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C-$$\end{document}numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range Wmax(Σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_\text {max}(\varSigma )$$\end{document} of a bounded set Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varSigma$$\end{document} of n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document} nonnegative matrices is introduced and some of its properties are also investigated.


Introduction and preliminaries
The algebraic system max algebra and its isomorphic versions (max plus algebra, 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 optimization, 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 nonlinear 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 = (a ij ) ∈ M m×n (ℝ), we say that A is positive (nonnegative) and write A > 0 (A ≥ 0) if a ij > 0 (a ij ≥ 0) for 1 ≤ i ≤ m , 1 ≤ j ≤ n. Let ℝ + be the set of all nonnegative real numbers and M m×n (ℝ + ) be the set of all m × n nonnegative real matrices. The notions M n (ℝ + ) and ℝ n + are considered for M n×n (ℝ + ) and M n×1 (ℝ + ), respectively. Let A = (a ij ) ∈ M m×n (ℝ + ) and B = (b ij ) ∈ M n×l (ℝ + ). Let r x (A) denote the local spectral radius of A, i.e., r x (A) = lim sup j→∞ ‖A j ⊗ ⊗ x‖ 1∕j . It was shown in [11] for x = [x 1 , … , x n ] t ∈ ℝ n + , x ≠ 0 that r x (A) = lim j→∞ ‖A j ⊗ ⊗ x‖ 1∕j and that r x (A) = max{r e i (A) ∶ i = 1, … , n, x i ≠ 0} , where e i denotes the ith standard basis vector and x i denotes the ith coordinate of x. We say that ≥ 0 is a geometric max eigenvalue of A if A ⊗ x = x for some x ≠ 0 , x ≥ 0 . Let max (A) denote the set of geometric max eigenvalues of A. The following result of Gunawardena was restated and reproved in [11,Theorem 2.7].

Theorem 1 If A ∈ M n (ℝ + ) , then
We define the standard vector multiplicity of geometric max eigenvalue as the number of indices j such that = r e j (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 ) associated to A is defined by setting N(A) = {1, … , n} and letting (i, j) ∈ E(A) whenever a ij > 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 ∈ M n (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,…,n r e j (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 S n is the group of permutations on {1, … , n} . The characteristic maxpolynomial 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 nondifferentiability 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 max (A) ⊂ trop (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 ∈ M n (ℝ + ) . The max-numerical range W max (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 trop (A) ⊂ W max (A) [15,Theorem 3.10]. It was proved in [15,Theorem 3.7] that given A ∈ M n (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 maxnumerical range and to provide interesting results for these generalizations.
As it will be evident from below the article is partly expository and is organized 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 (A 1 , … , A k ), where A i ∈ M n , i = 1, … , k, and we apply Theorem 2 to obtain a new formula for max joint numerical range W max ( ) of a bounded set of n × n nonnegative matrices (11). We move on in Section 4 to introduce some definitions and facts, which we need in our proofs and study the max k−numerical range W k max (A) , where k ≤ n is a positive integer. We explicitly describe a formula for W k max (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 ∈ M n (ℝ + ) . In the last section we introduce and study the max c−numerical range and max C−numerical range of nonnegative matrices, where c ∈ ℝ n + and C ∈ M n (ℝ + ). Also, we investigate some basic algebraic and geometrical properties of these sets.

Max-numerical range
Let M n (ℂ) be the vector space of all n × n complex matrices. The numerical range of a square matrix A ∈ M n (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 ∈ M n (ℝ + ) be a non-negative matrix. The max numerical range W max (A) of A is defined by It's obvious that i.e., for all 1 ≤ i ≤ n, 0 ≤ x i ≤ 1 and x j = 1 for some 1 ≤ j ≤ n.
(ii) Suppose that A ∈ M n (ℝ + ) and f A ∶ S ⟶ ℝ + , where So W max (A) is the image of the continuous function f A . Since S is a connected set, also W max (A) is a connected set.
To prove the reverse inclusion, recall that the function x ⟼ f A (x) = x t ⊗ A ⊗ x is continuous on the compact connected set ( Fig. 1) 1 ≤ k, r, s ≤ n. Now we consider four cases.
which completes the proof.
Since the maximum of differentiable functions is locally Lipschitz continuous, the following proposition follows.
In conventional algebra, a matrix U ∈ M n is called unitary if U * U = UU * = I n . By analogy one can make the following definition in max algebra: then U is called unitary in max algebra and we denote The following result was established in [3].

Proposition 2 Let A ∈ M n (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 ∈ M n (ℝ + ) be nonnegative matrices and let P be a permutation matrix. Then
For X, Y ⊆ ℝ + , recall that X ⊕ Y is defined as follows: In the following two results we collect some properties of W max . 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 ∈ M n (ℝ + ) be nonnegative matrices and let , ∈ ℝ + . Then the following statements hold.
The equality holds, when min It turns out that in several cases the quotients So the limit lim 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 following: if A ∈ M n (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 ultimately 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 ∈ M n (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 ∈ M n (ℝ + ) is irreducible, then there exist natural numbers p and T such that l(W max (A In the following, we consider two special cases when (8) holds for p = 1.

(i) If A is an upper triangular matrix or a lower triangular matrix, then
(ii) If max 1≤i,j≤n a ij = max 1≤i≤n a ii and the limit lim

exists, then it is equal
to the maximum of a ii on 1 ≤ i ≤ n.
Proof (i) By Propositions 3 and 4(v) we may assume without loss of generality that a 11 ≤ a 22 ≤ … ≤ a nn and that A is upper triangular matrix. By computing A m ⊗ , one can see that there exists some s ≥ n such that We claim that there exist 1 ≤ i 0 ≤ n and s 0 ≥ n such that If this is not the case, then for some large enough m ≥ n, there exist 1 ≤ i 1 , j 1 ≤ n such that This shows that which leads to a contradiction. This shows that the claim is true and the result follows since the minimal element on the diagonal of A m ⊗ is strictly smaller than a m nn for all m ≥ s 0 .
(ii) We may assume that a 11 ≤ a 22 ≤ ⋯ ≤ a nn . As max 1≤i,j≤n a ij = max 1≤i≤n a ii = a nn , it follows that It now follows from (9) that which completes the proof. ◻

Max joint numerical ranges
Recall that the joint numerical range of a k-tuple So, one can define the max joint numerical range of a k-tuple of n × n nonnegative matrices = (A 1 , … , A k ) in the following way:

Remark 3
Note that it follows from the above definition that From Theorem 2 and Remark 3 we conclude the following result.
Next we define a max joint numerical range W max ( ) 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]) The max Berger Wang formula asserts that ( [9, 10, 13]) where ‖A‖ = max i,j=1,…,n a ij (since all norms on ℝ n×n are equivalent one can use here any norm on ℝ n×n ). We also have ( ) = (S( )) , ‖ ‖ = sup 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 I k denote the k × k identity matrix. A matrix X ∈ M n×k (ℝ + ) is called an isometry in max algebra if X t ⊗ X = I k , and the set of all n × k isometry matrices in max algebra is denoted by X n×k . For the case k = n, X n×n is equal to U n , which was introduced in Definition 2.

Definition 3
For A ∈ M n (ℝ + ) with k ≤ n, the max k−numerical range of A is defined and denoted by It is clear that W 1 max (A) = W max (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 x i , x j ∈ ℝ n + ,

Remark 4
For a nonnegative matrix A = (a ij ) and 1 ≤ k ≤ n , the map f A ∶ X n×k ⟶ ℝ + is locally Lipschitz continuous on X n×k , where Note that W k max (A) is the image of the continuous function f A . Using connectivity and compactness of X n×k , W k max (A) is a connected and compact set.
We have the following explicit formula for W k max (A).

Theorem 3
Suppose that A = (a ij ) ∈ M n (ℝ + ) and let 1 ≤ k ≤ n be a positive integer. We have Proof By Definition 3 and Theorem 2, it follows that min W k max (A) = c and max W k max (A) = d . Since W max is a connected set, by Remark 4, this implies (13). ◻ In the following theorem, we state some basic properties of the max k−numerical range of matrices.

Theorem 4 Let
A ∈ M n (ℝ + ) and 1 ≤ k ≤ n . Then the following assertions hold: , and the equality holds if s = n; The properties (i), (ii), (iii) and (iv) easily follow from Theorem 3 (or directly from Definition 3).
To prove (v), let z ∈ W k+1 max (A) be given. So, by Definition 3, there exist Now, one can assume that, without loss of generality, Hence, by setting X = [x i 2 , … , x i k , x i k+1 ], we have X ∈ X n×k and so This implies that z ∈ W k max (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 spectrum and k-tropical spectrum of A ∈ M n (ℝ + ).  k

Definition 4 Let
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 , the following result follows from Definitions 4, 5 and 3 and Theorems 2 and 3.

Proposition 7 Let
By Theorem 3, (14) and (15), we have the following two results. In the following result we state the relationship between the max k−geometric spectrums of A.

Proposition 8 Let
Proposition 10 Let A ∈ M n (ℝ + ) and 1 ≤ k < n . Then Proof At first, let z ∈ k+1 max (A) be given. By Definition 4 there exists , the proof is complete. ◻ The following example from [11] illustrates the result above. The following analogue of Proposition 10 for the k−tropical spectrum is proved in a similar way as Proposition 10.

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 article and leave this for further research.

Max c-numerical range and max C-numerical range
Let A, C ∈ M n (ℝ + ) and c ∈ ℝ n + . Next we define and study max c-numerical range and max C-numerical range of A ∈ M n (ℝ + ) . To access more information about some known results in the complex case, see [8,Chapter1].

Definition 6
Let A ∈ M n (ℝ + ) and c = [c 1 , c 2 , … , c n ] t ∈ ℝ n + . The max c−numerical range of A is defined and denoted by In Definition 6, it's obvious that X ∈ U n , where the notation U n was denoted in (2).

Remark 7
Let A ∈ M n (ℝ + ) and c = [ , … , ] t ∈ ℝ n + . Then Since tr ⊗ (X t ⊗ A ⊗ X) = tr ⊗ (A) for X ∈ U n it follows that Next we introduce and study the notion of max C−numerical range of nonnegative matrices, where C ∈ M n (ℝ + ).

Definition 7
Let A, C ∈ M n (ℝ + ). The max C−numerical range of A is defined and denoted by Example 5 Let C = (c ij ) ∈ M n (ℝ + ) such that c 11 = 1 and c ij = 0 elsewhere. Then one can easily obtain that In the following theorem, we state some basic properties of the max C−numerical range of non-negative matrices.
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