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In this paper, we examine almost (strictly) semimonotone matrices which are also either Z -matrices or matrices having Property (+ +) and show that both parts of Wendler's conjecture [22] (2019) hold for these matrices. Another important result of this paper is the characterization of copositive matrices beyond the matrices with non-positive off-diagonal elements. We discuss some interesting properties related to the structure of almost (strictly) semimonotone matrices and also present results pertaining to the existence and multiplicity of solutions to the linear complementarity problem associated with an almost strictly semimonotone matrix. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a generalization of Schrödinger operator, known as Hamiltonian with finite range. We prove the related Gordon's lemma with quantitative conditions which means the absence of point spectrum. [ABSTRACT FROM AUTHOR]

Let G be a group and F an infinite field. Assume that A is a finite dimensional F -algebra with an elementary G -grading. In this paper, we study the graded identities satisfied by the tensor product grading on the F -algebra A ⊗ C , where C is an H -graded colour β -commutative algebra. More precisely, under a technical condition, we provide a basis for the T G -ideal of graded polynomial identities of A ⊗ C , up to graded monomial identities. Furthermore, the F -algebra of upper block-triangular matrices U T (d 1 , ... , d n) , as well as the matrix algebra M n (F) , with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤ 2 d − 1 , for M d (E) and M q (F) ⊗ U T (d 1 , ... , d n) , with a tensor product grading, is exhibited. In this latter case, d = d 1 + ... + d n. Here E denotes the infinite dimensional Grassmann algebra with its natural Z 2 -grading, and the grading on M q (F) is Pauli grading. The results presented in this paper generalize results from [14] and from other papers which were obtained for fields of characteristic zero. [ABSTRACT FROM AUTHOR]

For a Rirodan array, its vertical half and horizontal half are studied separately before. In this paper, we propose the (m 1 , s 1 , r , m 2 , s 2 , l) -halves of a 3-dimensional Riordan array, which are natural generalizations of the skew halves of a usual Riordan array in [21]. Let G = [ g i , j , k ] i , j , k ≥ 0 be a 3-dimensional Riordan array, its (m 1 , s 1 , r , m 2 , s 2 , l) -halves H = [ h i , j , k ] i , j , k ≥ 0 are defined by h i , j , k = g (m 1 + 1) i + (s 1 − 2) j + r , m 1 i + (s 1 − 1) j + r , m 2 i + (s 2 − 1) j + l k. We obtain that the (m 1 , s 1 , r , m 2 , s 2 , l) -halves of a 3-dimensional Riordan array are all 3-dimensional Riordan arrays. We also consider the (m 1 , s 1 , r , m 2 , s 2 , l) -antecedents of a 3-dimensional Riordan array. [ABSTRACT FROM AUTHOR]

Weighted shifts on directed trees are a decade-old generalisation of classical shift operators in the sequence space ℓ 2. In this paper we introduce the joint backward extension property (JBEP) for classes of weighted shifts on directed trees. If a class satisfies JBEP, the existence of a common backward extension within the class for a family of weighted shifts on rooted directed trees does not depend on the additional structure of the big tree (of fixed depth). We decide whether several classes of operators have JBEP. For subnormal or power hyponormal weighted shifts the property is satisfied, while it fails for completely hyperexpansive or quasinormal. Nevertheless, some positive results on joint backward extensions of completely hyperexpansive weighted shifts are proven. [ABSTRACT FROM AUTHOR]

Tutte's Matrix-Tree Theorem gives the number of spanning trees in a connected digraph by the determinant of a submatrix of the Laplacian, which is a generalization of the original Kirchhoff's Matrix-Tree Theorem associated to undirected graphs. In this paper, we obtain a new extension of Tutte's Matrix-Tree Theorem in terms of the signless Laplaicians. Moreover, we provide an elementary proof in terms of the factorization of signless Laplacian matrix into two incidence matrices. We finally generalize our result to the directed weighted graphs. [ABSTRACT FROM AUTHOR]

In this paper we propose some generalizations of real power form for Young-type inequalities which are much better than the recent results developed by many researchers. Our approaching method, together with idea on linear interpolation of Choi et al. (2017) [3] , allows us to build very strict improvements of real power form for Young-type inequalities. As applications, we give the operator versions and inequalities involving unitarily invariant norms and determinants of matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a one-parameter generalization of the famous Böttcher-Wenzel (BW) inequality in terms of a q -deformed commutator. For n × n matrices A and B , we consider the inequality Re 〈 [ B , A ] , [ B , A ] q 〉 ≤ c (q) ‖ A ‖ 2 ‖ B ‖ 2 , where 〈 A , B 〉 = tr (A ⁎ B) is the Hilbert-Schmidt inner product, ‖ A ‖ is the Frobenius norm, [ A , B ] = A B − B A is the commutator, and [ A , B ] q = A B − q B A is the q -deformed commutator. We prove that when n = 2 , or when A is normal with any size n , the optimal bound is given by c (q) = (1 + q) + 2 (1 + q 2) 2. We conjecture that this is also true for any matrices, and this conjecture is perfectly supported for n up to 15 by numerical optimization. When q = 1 , this inequality is exactly the BW inequality. When q = 0 , this inequality leads the sharp bound for the r -function which is recently derived for the application to universal constraints of relaxation rates in open quantum dynamics. [ABSTRACT FROM AUTHOR]

For two n × n complex matrices A and B , we define the q -deformed commutator as [ A , B ] q : = A B − q B A for a real parameter q. In this paper, we investigate a generalization of the Böttcher-Wenzel inequality which gives the sharp upper bound of the (Frobenius) norm of the commutator. In our generalisation, we investigate sharp upper bounds on the q -deformed commutator. This generalization can be studied in two different scenarios: firstly bounds for general matrices, and secondly for traceless matrices. For both scenarios, partial answers and conjectures are given for positive and negative q. In particular, denoting the Frobenius norm by | |. | | F , when A or B is normal, we prove the following inequality to be true and sharp: | | [ A , B ] q | | F 2 ≤ (1 + q 2) | | A | | F 2 | | B | | F 2 for positive q. Also, we conjecture that the same bound is true for positive q when A or B is traceless. For negative q , we conjecture other sharp upper bounds to be true for the generic scenarios and the scenario when A or B is traceless. All conjectures are supported with numerics and proved for n = 2. [ABSTRACT FROM AUTHOR]

The concept of coherent configurations is a generalization of the concept of association schemes. In this paper, we introduce the concept of Q -polynomial coherent configurations and study the relationship among intersection numbers, Krein numbers, and eigenmatrices. The examples of Q -polynomial coherent configurations are provided from Delsarte designs in Q -polynomial schemes and spherical designs, Euclidean designs. [ABSTRACT FROM AUTHOR]

The α -normal labelling method, which was introduced by Lu and Man in 2014, is an effective method for comparing the adjacency spectral radii of uniform hypergraphs. Recently, the α spectra of graphs and hypergraphs have attracted much attention from researchers. In this paper, we present a new kind of generalization of α -normal labelling method relating to the α -spectral radius of k -uniform hypergraphs. The effect of several types of graph operations on the α -spectral radius of the hypergraph are studied. As applications, we determine the unique hypergraph with the minimum α -spectral radius among connected k -uniform hypergraphs with given edge number, among c -cyclic hypergraphs with c ≥ 1 , respectively, and determine the unique supertree with second minimum α -spectral radius. Furthermore, we extend a recent result obtained by Wang and Yuan on the minimum spectral radius of supertrees with some constraints on maximal degree. [ABSTRACT FROM AUTHOR]

The authors of this paper have used the theory of Riordan matrices to introduce the notion of a Riordan graph in [3]. Riordan graphs are proved to have a number of interesting (fractal) properties, and they are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The main focus in [3] is the study of structural properties of families of Riordan graphs obtained from certain infinite Riordan graphs. In this paper, we use a number of results in [3] to study spectral properties of Riordan graphs. Our studies include, but are not limited to the spectral graph invariants for Riordan graphs such as the adjacency eigenvalues, (signless) Laplacian eigenvalues, nullity, positive and negative inertia indices, and rank. We also study determinants of Riordan graphs, in particular, giving results about determinants of Catalan graphs. [ABSTRACT FROM AUTHOR]

For an eigenvalue λ 0 of a Hermitian matrix A , the formula of Thompson and McEnteggert gives an explicit expression of the adjugate of λ 0 I − A , Adj (λ 0 I − A) , in terms of eigenvectors of A for λ 0 and all its eigenvalues. In this paper Thompson-McEnteggert's formula is generalized to include any matrix with entries in an arbitrary field. In addition, for any nonsingular matrix A , a formula for the elementary divisors of Adj (A) is provided in terms of those of A. Finally, a generalization of the eigenvalue-eigenvector identity and three applications of the Thompson-McEnteggert's formula are presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the class of diagonally dominant with respect to a given LMI region D ⊂ C matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region D. Moreover, we show that in some cases, diagonal D -dominance implies (D , D) -stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix). [ABSTRACT FROM AUTHOR]

Given a d -dimensional tensor T : A 1 × ... × A d → F (where F is a field), the i-flattening rank of T is the rank of the matrix whose rows are indexed by A i , columns are indexed by B i = A 1 × ... × A i − 1 × A i + 1 × ... × A d and whose entries are given by the corresponding values of T. The max-flattening rank of T is defined as mfrank (T) = max i ∈ [ d ] ⁡ frank i (T). A tensor T : A d → F is called semi-diagonal , if T (a , ... , a) ≠ 0 for every a ∈ A , and T (a 1 , ... , a d) = 0 for every a 1 , ... , a d ∈ A that are all distinct. In this paper we prove that if T : A d → F is semi-diagonal, then mfrank (T) ≥ | A | d − 1 , and this bound is the best possible. We give several applications of this result, including a generalization of the celebrated Frankl-Wilson theorem on forbidden intersections. Also, addressing a conjecture of Aharoni and Berger, we show that if the edges of an r -uniform multi-hypergraph H are colored with z colors such that each color class is a matching of size t , then H contains a rainbow matching of size t provided z > (t − 1) ( r t r ). This improves previous results of Alon and Glebov, Sudakov, and Szabó. [ABSTRACT FROM AUTHOR]

The desirable properties when constructing collections of subspaces often include the algebraic constraint that the projections onto the subspaces yield a resolution of the identity like the projections onto lines spanned by vectors of an orthonormal basis (the so-called tightness condition) and the geometric constraint that the subspaces form an optimal packing of the Grassmannian, again like the one-dimensional subspaces spanned by vectors in an orthonormal basis. In this article a generalization of related constructions which use known packings to build new configurations and which appear in numerous forms in the literature is given, as well as the characterization of a long list of desirable algebraic and geometric properties which the construction preserves. Another construction based on subspace complementation is similarly analyzed. While many papers on subspace packings focus only on so-called equiisoclinic or equichordal arrangements, attention is also given to other configurations like those which saturate the orthoplex bound and thus are optimal but lie outside of the parameter regime where equiisoclinic and equichordal packings can occur. [ABSTRACT FROM AUTHOR]

The theory of τ -summing and σ -nuclear linear operators on Banach spaces was developed by Pietsch [20, Chapter 23] . Extending the linear case to the range p > 1 and generalizing all cases to the multilinear setting, in this paper we introduce the concept of σ ( p ) -nuclear linear and multilinear operators. In order to develop the duality theory for the spaces of such operators, we introduce the concept of quasi- τ ( p ) -summing linear/multilinear operators and prove Pietsch-type domination theorems for such operators. The main result of the paper shows that, under usual conditions, linear functionals on the space of σ ( p ) -nuclear n -linear operators are represented, via the Borel transform, by quasi- τ ( p ) -summing n -linear operators. As far as we know, this result is new even in the linear case n = 1 . [ABSTRACT FROM AUTHOR]

In this paper three new classes of matrices that have a computationally simple inverse are studied. Specifically, semi-involutory matrices are characterized for small orders and many interesting results are provided. Semi-involutory matrices can be thought of as a generalization of involutory matrices. In addition, signed semi-involutory matrices are studied and a natural generalization is introduced. [ABSTRACT FROM AUTHOR]

In this paper, we generalize and analyze the concepts of diagonal dominance and irreducibility in the framework of column and row representative matrices of a set. Our analysis includes the definition of particular sets of M - and H -matrices. We also analyze the form that the matrices of the introduced irreducible sets must have and the implications of the obtained results on the solution of vertical and horizontal linear complementarity problems. In this context, we prove that the projected Jacobi and the projected Gauss-Seidel methods for horizontal linear complementarity problems converge when the matrices of the problem satisfy one of the introduced generalizations of strict diagonal dominance or of irreducible diagonal dominance. This extends current convergence results. [ABSTRACT FROM AUTHOR]

In this paper we analyze the nearly optimal block diagonal scalings of the rows of one factor and the columns of the other factor in the triangular form of the SR decomposition. The result is a block generalization of the result of the van der Sluis about the almost optimal diagonal scalings of the general rectangular matrices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the free sets and the free subsemimodules in a semimodule over a commutative semiring S . First, we discuss some properties of the free sets and give a sufficient condition for a nonempty finite set to be free in a finitely generated free S -semimodule and obtain a relation between free set and linear independent set in an S -semimodule. Then we consider the free subsemimodules and prove that the rank of any free subsemimodule of a finitely generated S -semimodule M does not exceed that of M . Also, we give some equivalent descriptions for a commutative semiring S to have the property that all nonzero subsemimodules of any finitely generated free S -semimodule are free. Partial results obtained in the paper develop and generalize the corresponding results for modules over rings and linear spaces over fields. [ABSTRACT FROM AUTHOR]

The paper by I. Hnětynková et al. (2015) [11] introduces real wedge-shaped matrices that can be seen as a generalization of Jacobi matrices, and investigates their basic properties. They are used in the analysis of the behavior of a Krylov subspace method: The band (or block) generalization of the Golub–Kahan bidiagonalization. Wedge-shaped matrices can be linked also to the band (or block) Lanczos method. In this paper, we introduce a complex generalization of wedge-shaped matrices and show some further spectral properties, complementing the already known ones. We focus in particular on nonzero components of eigenvectors. [ABSTRACT FROM AUTHOR]

In 2007, Yuan and Leng gave a generalization of Ky Fan's determinantal inequality, which is a refinement of the fundamental Brunn-Minkowski inequality (det ⁡ (A + B)) 1 / n ≥ (det ⁡ A) 1 / n + (det ⁡ B) 1 / n , where A and B are positive semidefinite matrices. In this paper, we first give an extension of Yuan-Leng's result to multiple positive definite matrices, and we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector. Our result improves a recent result of Liu (2016) [16]. [ABSTRACT FROM AUTHOR]

This paper is concerned with a generalization of the concept of value of a (zero-sum) matrix game. Given a finite dimensional real inner product space V with a self-dual cone K , an element e in the interior of K , and a linear transformation L , we define the value of L by v ( L ) : = max x ∈ Δ ⁡ min y ∈ Δ ⁡ 〈 L ( x ) , y 〉 = min y ∈ Δ ⁡ max x ∈ Δ ⁡ 〈 L ( x ) , y 〉 , where Δ = { x ∈ K : 〈 x , e 〉 = 1 } . This reduces to the classical value of a square matrix when V = R n , K = R + n , and e is the vector of ones. In this paper, we extend some classical results of Kaplansky and Raghavan to this general setting. In addition, for a Z -transformation (which is a generalization of a Z -matrix), we relate the value with various properties such as the positive stable property, the S -property, etc. We apply these results to find the values of the Lyapunov transformation L A and the Stein transformation S A on the cone of n × n real symmetric positive semidefinite matrices. [ABSTRACT FROM AUTHOR]

In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed several conjectures in 2016. In this paper we discuss further these conjectures and put forward a couple of new conjectures which will be shown equivalent to one of the conjectures. In particular, we prove this conjecture and hence all conjectures in some special cases. Another conjecture is shown with a slightly bigger upper bound. In addition, the developed methods give some new simple proofs of the complex BW inequality and its condition for equality. [ABSTRACT FROM AUTHOR]

In this paper we consider two polynomial chains and we study the existence of an intermediate polynomial chain satisfying interlacing inequality relations together with some convexity-like properties that arise from applications in completion problems. The presented result is a vast generalisation of existing results dealing with intermediate polynomial chains. The proof of the main result is completely novel to the field, different from all the existing methods used in similar problems. The obtained necessary and sufficient conditions are simple, explicit and constructive. [ABSTRACT FROM AUTHOR]

Recently, Liu (2016) [9] obtained some determinantal inequalities for sector matrices. In this paper, we make considerable improvement on Liu's results. [ABSTRACT FROM AUTHOR]

Abstract: The study of m-isometry elementary operator of length one is initiated by Botelho and Jamison. Several authors have contributed to obtain this result: if the two operator coefficients involved are m-isometries then the elementary operator is m-isometry on the Hilbert–Schmidt operator ideal. In this paper we show that the converse of this result is also true. We also give a necessary and sufficient condition for the generalized derivation to be an m-isometry. The sufficiency follows from results on the sum of m-isometries and nilpotent operators in two recent papers. [Copyright &y& Elsevier]

Abstract: New families of classical and quantum optimal negacyclic convolutional codes are constructed in this paper. This optimality is in the sense that they attain the classical (quantum) generalized Singleton bound. The constructions presented in this paper are performed algebraically and not by computational search. [Copyright &y& Elsevier]

Abstract: In this paper, the bases of a semimodule over a commutative semiring R are investigated. Some properties and characterizations of the bases are discussed and some equivalent conditions for a basis to be a free basis in a finitely generated free semimodule over R are given. The different possible cardinalities for a basis in a finitely generated free semimodule over R are considered and some equivalent descriptions are obtained for a commutative semiring R satisfying the property that any two bases for a finitely generated free R-semimodule have the same cardinality. Partial results obtained in the paper develop and generalize the corresponding results for commutative join-semirings and for commutative zerosumfree semirings. [Copyright &y& Elsevier]

Abstract: In two recent papers by Barry (2010) and (2011) , it is conjectured that Somos-4 admits a solution expressed in terms of Hankel determinant with its elements satisfying a convolution recursion relation. In this paper, Barry’s conjecture on Somos-4 is firstly confirmed. Actually, we present a more generalized result. The proof is mainly based on new findings on properties for so-called Block–Hankel determinants. The method can also be used to prove another conjecture proposed by Michael Somos, which has been solved by Guoce Xin. [Copyright &y& Elsevier]

The signature s (G) of a graph G is defined as the difference between its positive inertia index and negative inertia index. In 2013, H. Ma et al. ([1]) conjectured that − c 3 (G) ≤ s (G) ≤ c 5 (G) for an arbitrary simple graph G , where c 3 (G) denotes the number of cycles in G of length 3 modulo 4, c 5 (G) denotes the number of cycles in G of length 1 modulo 4. In this paper, we prove that this conjecture holds for claw-free graphs and graphs with least eigenvalue ≥−2. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to a generalization of a Hadamard type inequality for the permanent of a complex square matrix. Our proof is based on a non-trivial extension of a technique used in Carlen et al. (2006) [5]. We give an application to coefficients of products of linear forms and show some auxiliary inequalities, which might be of independent interest. [ABSTRACT FROM AUTHOR]

Abstract Linearization is a widely used method for solving polynomial eigenproblems in which a matrix polynomial is transformed to a matrix pencil of larger size. Fiedler pencils, generalized Fiedler pencils, Fiedler pencils with repetition (FPRs) and generalized Fiedler pencils with repetition (GFPRs) are well known classes of strong linearizations of matrix polynomials. The class GFPRs is also a rich source of structure-preserving strong linearizations of structured matrix polynomials. The recovery of eigenvectors, minimal bases and minimal indices of matrix polynomials from those of the linearizations is an important task. It is well known that eigenvectors and minimal bases of matrix polynomials can be easily recovered from those of the Fiedler and generalized Fiedler pencils. By contrast, barring a small subclass of FPRs, the recovery of eigenvectors and minimal bases of matrix polynomials from those of the FPRs and GFPRs is still an open problem. The purpose of this paper is to fill this gap. We consider the class of GFPRs, which subsumes the class of FPRs, and describe the recovery of eigenvectors, minimal bases and minimal indices of matrix polynomials from those of the GFPRs and show that the recovery is operation-free. [ABSTRACT FROM AUTHOR]

Abstract A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex u ∈ V a weight that equals the corresponding entry ν u of the Perron eigenvector ν . This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity by using a Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular graphs. In addition, we show an application of weight-regular partitions to study graphs that attain equality in the classical Hoffman's lower bound for the chromatic number of a graph, and we show that weight-regularity provides a condition under which Hoffman's bound can be improved. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a simple undirected graph with the characteristic polynomial of its Laplacian matrix , . Aleksandar Ilić [A. Ilić, Trees with minimal Laplacian coefficients, Comput. Math. Appl. 59 (2010) 2776–2783] identified n-vertex trees with given matching number q which simultaneously minimize all Laplacian coefficients. In this paper, we give another proof of this result. Generalizing the approach in the above paper, we determine n-vertex trees with given matching number q which have the second minimal Laplacian coefficients. We also identify the n-vertex trees with a perfect matching having the largest and the second largest Laplacian coefficients, respectively. Extremal values on some indices, such as Wiener index, modified hyper-Wiener index, Laplacian-like energy, incidence energy, of n-vertex trees with matching number q are obtained in this paper. [Copyright &y& Elsevier]

Abstract Let V be a Euclidean Jordan algebra of rank n. A set E in V is said to be a spectral set if there exists a permutation invariant set Q in R n such that E = λ − 1 (Q) , where λ : V → R n is the eigenvalue map that takes x ∈ V to λ (x) (the vector of eigenvalues of x written in the decreasing order). If the above Q is also a convex cone, we say that E is a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit [ x ] : = { y : λ (y) = λ (x) } is arcwise connected, we show that if a permutation invariant set Q is connected (arcwise connected), then λ − 1 (Q) is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible. [ABSTRACT FROM AUTHOR]

An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved. [ABSTRACT FROM AUTHOR]

We focus on the periodicity of the Grover walk on the generalized Bethe tree, which is a rooted tree such that in each level the vertices have the same degree. Since the Grover walk is induced by the underlying graph, its properties depend on the graph. In this paper, we say that the graph induces periodic Grover walks if and only if there exists k ∈ N such that the k -th power of the time evolution operator becomes the identity operator. Our aim is to characterize such graphs. We give the perfect characterizations of the generalized Bethe trees which induce periodic Grover walks. [ABSTRACT FROM AUTHOR]

In this paper, we develop a polynomial time algorithm to compute a Dulmage–Mendelsohn-type decomposition of a matrix partitioned into submatrices of rank at most 1. [ABSTRACT FROM AUTHOR]

The paper concerns generalizations of the Böttcher–Wenzel inequality to contracted products of tensors. We show that the best constant in the inequality is as expected in some cases and present an example where the best constant is larger than expected. [ABSTRACT FROM AUTHOR]

In this paper we characterize additive maps between tensor spaces that send decomposable tensors to decomposable tensors. As an application, we classify all additive maps from tensor products of spaces of rectangular matrices to spaces of rectangular matrices which do not increase the rank of tensor product of rank one matrices. [ABSTRACT FROM AUTHOR]

We extend to hypermatrices definitions and theorem from matrix theory. Our main result is an elementary derivation of the spectral decomposition of hypermatrices generated by arbitrary combinations of Kronecker products and direct sums of cubic side length 2 hypermatrices. The method is based on a generalization of Parseval's identity. We use this general formulation of Parseval's identity to introduce hypermatrix Fourier transforms and discrete Fourier hypermatrices. We extend to hypermatrices a variant of the Gram–Schmidt orthogonalization process as well as Sylvester's classical Hadamard matrix construction. We conclude the paper with illustrations of spectral decompositions of adjacency hypermatrices of finite groups and a short proof of the hypermatrix formulation of the Rayleigh quotient inequality. [ABSTRACT FROM AUTHOR]

An ordered pair of two adjacent vertices in a graph is called an arc. In this paper, we give the determinant and the inverse, whenever it exists, of the distance matrix of an arc-weighted tree with arc weights in a ring. These generalize known results. We also give examples that cannot be dealt with by known results. However, our results can be applied for these examples. [ABSTRACT FROM AUTHOR]

Inspired by recent works on m -isometries for a positive integer m , in this paper we introduce the classes of ∞-isometries and ∞-unitaries on a Hilbert space. We show that an ∞-isometry on a finite dimensional complex Hilbert space H with dimension N is in fact an ( 2 N − 1 ) -isometry. We describe the spectra of such operators, study the quasinilpotent perturbations of ∞-isometries and characterize when tensor products of ∞-isometries are also ∞-isometries. As a surprising by-product, we obtain a generalization of Nagy–Foias–Langer decomposition of a contraction into an unitary and a completely nonunitary contraction. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to generalize matrix numerical ranges to the tensor case based on tensor norms. We show that the basic properties of matrix numerical ranges such as compactness and convexity are valid for tensor numerical ranges. We make use of convexity property to propose an algorithm for approximating tensor numerical ranges in which tensor eigenvalues are contained. Also we consider tensor numerical ranges based on inner products, however, they may not be convex in general. [ABSTRACT FROM AUTHOR]

It is known that the numerical radius of the Hadamard product A ∘ B of two n -by- n matrices A and B is related to those of A and B by (a) w ( A ∘ B ) ≤ 2 w ( A ) w ( B ) , (b) w ( A ∘ B ) ≤ w ( A ) w ( B ) if one of A and B is normal, and (c) w ( A ∘ B ) ≤ ( max i ⁡ a i i ) w ( B ) if A = [ a i j ] i , j = 1 n is positive semidefinite. In this paper, we give complete characterizations of A and B for which the equality is attained. The matrices involved can be considered as elaborate generalizations of the equality-attaining A = [ 0 0 a 0 ] and B = [ 0 0 b 0 ] for (a), A = [ a 1 0 0 a 2 ] ( | a 1 | ≥ | a 2 | ) and B = [ w ( B ) ⁎ ⁎ ⁎ ] for (b), and A = [ a 1 a 3 a 2 a 4 ] ≥ 0 ( a 1 ≥ a 4 ) and B = [ w ( B ) ⁎ ⁎ ⁎ ] for (c). [ABSTRACT FROM AUTHOR]

We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are a decomposition of the eigenscheme of a matrix into primary components and the fact that this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint. [ABSTRACT FROM AUTHOR]

In this paper, we describe the eigenstructure and the Jordan form of the Fourier transform matrix generated by a primitive N -th root of unity in a field of characteristic 2. We find that the only eigenvalue is λ = 1 and its eigenspace has dimension [ N 4 ] + 1 ; we provide a basis of eigenvectors and a Jordan basis. The problem has already been solved, for number theoretic transforms, in any other finite characteristic. However, in characteristic 2 classical results about geometric multiplicity do not apply and we have to resort to different techniques in order to determine a basis of eigenvectors and a Jordan basis. We make use of a modified version of the Vandermonde's formula, which applies to matrices whose entries are powers of elements of the form x + x − 1 . [ABSTRACT FROM AUTHOR]

Given simple graphs G , H 1 , … , H n , where n = | V ( G ) | , the generalized corona, denoted G ∘ ˜ Λ i = 1 n H i , is the graph obtained by taking one copy of graphs G , H 1 , … , H n and joining the i th vertex of G to every vertex of H i . In this paper, we determine and study the characteristic, Laplacian and signless Laplacian polynomial of G ∘ ˜ Λ i = 1 n H i . This leads us to construct new pairs of cospectral, L -cospectral and Q -cospectral graphs. As an application, we give a simple proof for Csikvari's Lemma on eigenvalues of graphs. [ABSTRACT FROM AUTHOR]