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We introduce an equivalence relation ρ on matrices over a class of semirings, which is relevant to the Kleene star operator on matrices. We characterize those surjective linear transformations for matrices preserving ρ. Also, we show that a surjective linear transformation preserves Kleene stars of matrices if and only if it commutes with the Kleene star operator if and only if it preserves ρ. Finally, we illustrate that there exist non-surjective linear transformations that preserve ρ or preserve Kleene stars of matrices or commute with the Kleene star operator. [ABSTRACT FROM AUTHOR]

Let D be a division ring, let M be a right vector space over D and let End (M D ) be the ring of all D -linear transformations from M into M. Suppose that R is a dense subring of End (M D ) consisting of finite rank transformations and f : R → End (M D ) is an additive map. We show that if f (x) x m (x) = x m (x) f (x) for every rank-k transformation x ∈ R , where k is a fixed integer with 1 < k < dim M D and m (x) ≥ 1 is an integer depending on x, then there exist λ ∈ Z (D) and an additive map μ : R → Z (D) I such that f (x) = λ x + μ (x) for all x ∈ R , where I denotes the identity transformation on M. This gives a natural generalization of the recent results obtained in [Franca, Commuting maps on rank-k matrices. Linear Alg Appl. 2013;438:2813–2815; Liu and Yang, Power commuting additive maps on invertible or singular matrices. Linear Alg Appl. 2017;530:127–149] and can be regarded as an infinite-dimensional version of the Franca theorem obtained in [Franca, Commuting maps on rank-k matrices. Linear Alg Appl. 2013;438:2813–2815]. [ABSTRACT FROM AUTHOR]

A square matrix is said to be diagonalizable if it is similar to a diagonal matrix. In this paper, we discuss diagonability of matrices over commutative semirings and give an equivalent condition for an idempotent matrix over a commutative semiring to be diagonalizable. Also, we obtain an equivalent description for a matrix over a multiplicatively cancellative and commutative semiring to be diagonalizable. [ABSTRACT FROM AUTHOR]

Fillmore's Theorem says that if A is a nonscalar matrix of order n over a field F and γ 1 , γ 2 , ... , γ n ∈ F are such that γ 1 + γ 2 + ⋯ + γ n = t r A , then there is a matrix B of order n over F similar to A with diagonal (γ 1 , γ 2 , ... , γ n). Borobia extended Fillmore's Theorem to the matrices over the ring Z of integers. In this paper, we will extend Fillmore's Theorem to the matrices over a general factorial ring. [ABSTRACT FROM AUTHOR]

Let D m × n be the set of m × n matrices over a division ring D . Two matrices A , B ∈ D m × n are adjacent if rank (A − B) = 1. By the adjacency, D m × n is a connected graph. Suppose D , D ′ are division rings and m , n , m ′ , n ′ ≥ 2 are integers. When | D | ≥ 4 , we characterize the graph homomorphism ϕ : D n × n → D ′ m ′ × n ′ if ϕ (0) = 0 and there exists A 0 ∈ D n × n such that rank (ϕ (A 0)) = n. [ABSTRACT FROM AUTHOR]

Reverse-order law for the Moore–Penrose inverses of tensors is useful in the field of multilinear algebra. In this paper, we first prove some more identities involving the Moore–Penrose inverses of tensors. We then obtain a few necessary and sufficient conditions of the reverse-order law for the Moore–Penrose inverses of tensors via the Einstein product. [ABSTRACT FROM AUTHOR]

The edge-Zagreb matrix of a graph G whose vertex has degree is defined by if the vertices and are adjacent and otherwise. The edge-Zagreb spectral radius is the largest eigenvalue of the edge-Zagreb matrix of a graph G denoted by . The edge-Zagreb energy ZE(G) of the graph G is the sum of absolute values of the eigenvalues of its edge-Zagreb matrix. In this paper, we obtain several properties of the edge-Zagreb matrix and some upper bounds for , and characterize the extremal graphs. We also discuss upper and lower bounds of the edge-Zagreb energy ZE(G). [ABSTRACT FROM AUTHOR]

The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory. [ABSTRACT FROM AUTHOR]

The theory of Schur complement is very important in many fields such as control theory and computational mathematics. In this paper, by applying the properties of the Schur complement and some inequality techniques, some new estimates of the diagonally,-diagonally and product-diagonally dominant degree on the Schur complement of matrices are obtained, which improve some relative results. Further, as an application of these derived results, we present some distributions for the eigenvalues of the Schur complements. Finally, the numerical example is given to show the advantages of our derived results. [ABSTRACT FROM AUTHOR]

In 1958, Gerstenhaber showed that ifis a subspace of the vector space of the square matrices of ordernover some fieldconsisting of nilpotent matrices only (to be called anilspace) and if the underlying fieldis sufficiently large, then the maximal dimension ofis. This dimension is attained if and only if the linear space is similar to the space of all strictly upper-triangular matrices. In this paper, we study maximal spaces of nilpotent square matrices of ordern. As a striking extension of the Gerstenhaber’s result, we prove that a maximal nilspace(with the underlying fieldbeing sufficiently large) is similar to a (subspace of) all strictly upper-triangular matrices if and only if it contains a nilpotentJof maximal possible rankand its square. We give a twisted but elementary proof of this fact. [ABSTRACT FROM PUBLISHER]

A celebrated theorem of Shoda states that over any field(of characteristic ), every matrix with tracecan be expressed as a commutator, or, equivalently, that the set of values of the polynomialoncontains all matrices with trace. We generalize Shoda’s theorem by showing that every non-zero multilinear polynomial of degree at most, with coefficients in, has this property. We further conjecture that this holds for every non-zero multilinear polynomial with coefficients inof degree, provided that. [ABSTRACT FROM PUBLISHER]

Both the Gershgorin and Brauer eigenvalue inclusion sets reduce to a single disc when applied to a matrix with a constant main diagonal. We show that a union of ovals of Cassini containing all the eigenvalues of such a matrix can nonetheless be derived and that this union is guaranteed to lie inside the Brauer set. We then apply our results to Toeplitz matrices. [ABSTRACT FROM PUBLISHER]

In this article, we study the n × n matrices over a finite distributive lattice L. By using join irreducible elements in L, we first give some concrete ways to decompose L into a subdirect product of some chains. Also, it is showed that by a subdirect embedding from semiring R to the direct product of semirings R 1, R 2, … , R m , we can give a corresponding subdirect embedding from the matrix semiring M n (R) to semiring . Based on the above results, it is proved that a square matrix over a finite distributive lattice L can be decomposed into the sum of matrices over some special subchains of L. This generalizes and extends the corresponding results obtained by Fan [Z.T. Fan, The Theory and Applications of Fuzzy Matrices, Science Publication, Beijing, 2006 (in Chinese)] and by Zhao et al. [X.Z. Zhao, Y.B. Jun, and F. Ren, The semiring of matrices over a finite chain, Inform. Sci. 178 (2008), pp. 3443–3450]. As some applications, we present a method to calculate the indices and periods of the matrices over a finite distributive lattice, and characterize the idempotent and nilpotent matrices over a finite distributive lattice. Also, we discuss Green's relations on the multiplicative semigroup of semiring of matrices over a finite distributive lattice. [ABSTRACT FROM PUBLISHER]

A sign pattern matrix M with zero trace is primitive non-powerful if for some positive integer k, Mk = J#. The base l(M) of the primitive non-powerful matrix M is the smallest integer k. By considering the signed digraph S whose adjacent matrix is the primitive non-powerful matrix M, we will show that if l(M) = 2, the minimum number of non-zero entries of M is 5n - 8 or 5n - 7 depending on whether n is even or odd. [ABSTRACT FROM AUTHOR]

We derive an inclusion region for the eigenvalues of a matrix that can be considered an alternative to the Brauer set. It is accompanied by non-singularity conditions. [ABSTRACT FROM AUTHOR]

A 2-coloring (G1, G2) is primitive if there exist non-negative integers h and k with h + k > 0 such that for each pair (u, v) of vertices there exists an (h, k)-walk in G from u to v. The exponent of (G1, G2) is the minimum value of h + k taken over all such h and k. In this article, we consider 2-colorings of loopless, symmetric digraphs give the conditions for a 2-coloring of loopless, symmetric digraph to be primitive and establish an upper bound on the exponents. [ABSTRACT FROM AUTHOR]

The transversals of a complex square matrix and the isolated set of transversals were used in (McDonald, J.J., Olesky, D.D. and Tastsomeros, M.J., 1997, Ray patterns of matrices and nonsigularity. Linear Algebra and its Applications, 267, 359–373) to study the ray-nonsingular matrices and the determinantal regions of square ray pattern matrices. Some necessary conditions of the isolated sets of transversals were obtained in (McDonald, J.J., Olesky, D.D. and Tastsomeros, M.J., 1997, Ray patterns of matrices and nonsigularity. Linear Algebra and its Applications, 267, 359–373). A characterization of the isolated sets of transversals in an important special case was also given in (McDonald, J.J., Olesky, D.D. and Tastsomeros, M.J., 1997, Ray patterns of matrices and nonsigularity. Linear Algebra and its Applications, 267, 359–373). The study of the isolated set of transversals is indeed equivalent to the study of the isolated set of permutations on the set {1, .., n}. In this article, we give a graph theoretical characterization of the isolated set of permutations. We also give an application of this characterization in a problem concerning the linear independence of the set of permutation matrices contained in a given square (0, 1) matrix. [ABSTRACT FROM AUTHOR]

We studied the existence and uniqueness of solutions of the following two kinds of nonlinear matrix equations , and , and the convergence of the matrix sequences { X i } recursively defined by and where for k -1 and are initial positive definite matrices. [ABSTRACT FROM AUTHOR]

Using a simple concavity argument together with the scale-symmetries of the permanent function, we derive a handful of upper bounds for the permanent of a nonnegative matrix. Each is better than our previously best upper bound, and agrees with the Minc-Brègman bound when applied to a (0,1)-matrix. We show how sharpening , a different application of scale-symmetries, can be used to improve upon virtually any extant permanental upper or lower bound for nonnegative matrices. Indeed, when applied to a (0,1)-matrix, the sharpened versions of the new upper bounds typically improve, sometimes considerably, upon the Minc-Brègman bound itself. We also indicate how our methods apply to lower bounds. [ABSTRACT FROM AUTHOR]