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Let k and n be two coprime positive integers. In this paper, we show that any $ n\times n $ n × n k-circulant operator matrix $ A_{k,n} $ A k , n with the first row $ A_{1},\dots,A_{n}\in B(H) $ A 1 , ... , A n ∈ B (H) is unitarily equivalent to a generalized permutation operator matrix, an operator matrix that has at most one nonzero entry in each row and in each column. Among other applications in the subject of invertibility, eigenvalues and eigenvectors, using this unitary equivalence in the particular cases of $ k^{2}\pm 1=n $ k 2 ± 1 = n $ ({\rm mod} n) $ (mod n) , we give the following new formulas for computing the numerical radius of these operators: \[ w(A_{k,n})= \begin{cases} \frac{1}{2}\displaystyle \max_{1\leq i\leq n} \sup_{\theta \in \mathbb{R}} \left\| \begin{bmatrix} {\rm e}^{\mathrm{i}\theta}B_{i+1} & B^{*}_{ki+1}\\ B^{*}_{-ki+1} & {\rm e}^{\mathrm{i}\theta}B_{-i+1} \end{bmatrix} \right\|, & {\rm if} \ k^{2}+1\\ & \quad =n \ ({\rm mod}\,n)\\ \displaystyle\max_{1\leq i \leq n}w \left(\begin{bmatrix} 0 & B_{k(i-1)+1}\\ B_{i} & 0 \end{bmatrix} \right), & {\rm if} \ k^{2}-1\\ & \quad =n \ ({\rm mod}\,n), \end{cases} \] w (A k , n) = { 1 2 max 1 ≤ i ≤ n sup θ ∈ R ‖ [ e i θ B i + 1 B ki + 1 ∗ B − ki + 1 ∗ e i θ B − i + 1 ] ‖ , if k 2 + 1 = n (mod n) max 1 ≤ i ≤ n w ([ 0 B k (i − 1) + 1 B i 0 ]) , if k 2 − 1 = n (mod n) , where $ B_{j}=\sum _{s=1}^{n}\omega ^{(s-1)(j-1)}A_{s} $ B j = ∑ s = 1 n ω (s − 1) (j − 1) A s and $ \omega ={\rm e}^{\frac {2\pi \mathrm {i}}{n}} $ ω = e 2 π i n , the nth root of unity. [ABSTRACT FROM AUTHOR]

Vertices in the graph of a square matrix over a field may be classified as to how their removal changes the geometric multiplicity of an identified eigenvalue. There are three possibilities: $ +1 $ + 1 (Parter); no change (neutral); and $ -1 $ − 1 (downer). When the graph is a tree, the 'downer branch mechanism' distinguishes the Parter vertices. Here, we discover how this mechanism generalizes for general graphs, both for Hermitian matrices and general matrices. Then, we apply the new ideas, both to classify pendent edges in general graphs, and to understand the existence of 2-downer edge cycles in general graphs, when there is a 2-downer edge. This is a further explanation of why such edges cannot occur in trees. [ABSTRACT FROM AUTHOR]

For a tree T, U (T) denotes the minimum number of eigenvalues of multiplicity 1 among all real symmetric matrices, whose graph is T. It is known that U (T) ≥ 2. A tree is linear if all its vertices of degree at least 3 lie on a single induced path, and k-linear if there are k of these high degree vertices. U (T) is known for generalized stars (1-linear trees), and U (T) is determined here for 2-linear trees. Three (and higher) linear trees offer considerable additional complications. [ABSTRACT FROM AUTHOR]

In this paper, G-majorization inequalities are proven for the gradients of two Gateaux differentiable G-increasing functions defined on a real linear space endowed with the structure of an Eaton triple. Such inequalities can be interpreted as a monotonicity property of certain operators and functionals with respect to a special preorder on the class of all G-increasing functions. The notion of a c-strongly G-increasing function is introduced and used to illustrate the derived inequalities. The obtained results are also applied for matrix Eaton triples connected with the eigenvalues map on the space of Hermitian matrices as well as with the singular values map on the space of complex matrices. [ABSTRACT FROM AUTHOR]

A hypergraph is said to be oriented if each edge-vertex incidence has a label of +1 or −1. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and the other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G) = { v 1 , ... , v n } and E P (G) be an n × n matrix whose (i , j) -entry is the maximum number of internally edge-disjoint paths between v i and v j , if i ≠ j , and zero otherwise. Also, define E P ¯ (G) = E P (G) + D , where D is a diagonal matrix whose i-th diagonal element is the number of edge-disjoint cycles containing v i , whose E P (G) is a multiple of J−I. Among other results, we determine the spectrum and the energy of the matrix E P ¯ (G) for an arbitrary bicyclic graph G. [ABSTRACT FROM AUTHOR]

We present a characterization of eigenvalue inequalities between two Hermitian matrices by means of inertia indices. As applications, we deal with some classical eigenvalue inequalities for Hermitian matrices, including the Cauchy interlacing theorem and the Weyl inequality, in a simple and unified approach. We also give a common generalization of eigenvalue inequalities for (Hermitian) normalized Laplacian matrices of simple (signed, weighted, directed) graphs. Our approach is also suitable for Hermitian matrices of the second kind of digraphs recently introduced by Mohar. [ABSTRACT FROM AUTHOR]

The main goal of this article is to present several inequalities for the eigenvalues of several operators, when filtered through positive linear mappings, convex functions, superquadratic functions and asynchronous functions. [ABSTRACT FROM AUTHOR]

We derive a new family of polynomial multipliers that improve the Pellet radii of matrix polynomials, which determine an annulus in the complex plane devoid of eigenvalues, and that generalize and improve previously obtained results, while relaxing the conditions under which these hold. As a special limit case we also obtain an improvement of the Cauchy radius, which is an upper bound on the moduli of the eigenvalues. Results for the special case of scalar polynomials are presented. [ABSTRACT FROM AUTHOR]

Let G be the set of simple graphs (or multigraphs) G such that for each G ∈ G there exists at least two non-empty disjoint proper subsets V 1 , V 2 ⊆ V (G) satisfying V (G) ∖ (V 1 ∪ V 2) ≠ φ and edge connectivity κ ′ (G) = e (V i , V (G) ∖ V i) for 1 ≤ i ≤ 2. A multigraph is a graph with possible multiple edges, but no loops. Let τ (G) be the maximum number of edge-disjoint spanning trees of a graph G. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of τ (G) , we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bounds of κ ′ (G) and τ (G) of a simple graph or a multigraph G ∈ G , respectively. [ABSTRACT FROM AUTHOR]

This paper is a continuation of a series of papers on related topics by the authors. It is divided into two main parts. In the first part, the set of n × n e-matrices (real constant row-sum matrices) is partitioned into equivalence classes which are called e-similarity classes. The relationships between the spectra of different matrices belonging to the same equivalence class, as well as their left and generalized left eigenspaces, are discussed. In the second part, the results obtained in the first part are applied to improve the location of eigenvalues of e-matrices. Associated localization Gershgorin regions of the so-called second type are obtained. A number of examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we study multiplications and eigenvalues of tensors by using linear maps. By the composition of linear maps, we naturally obtain two multiplications of tensors, which generalize Shao's product, Einstein product and n-mode product etc. Some properties of product of tensors are obtained directly, which don't need proofs, for example the associative law of Shao's product etc. Moreover, we extend eigenvalues of tensors to tensor representations of quivers. By using tensor representations of quivers, eigenvalues of tensors and properties are given. [ABSTRACT FROM AUTHOR]

A fractional matching of a graph G is a function f giving each edge a number in [ 0 , 1 ] so that ∑ e ∈ Γ (v) f (e) ≤ 1 for each v ∈ V (G) , where Γ (v) is the set of edges incident to v. The fractional matching number of G, written ν ∗ (G) , is the maximum of ∑ e ∈ E (G) f (e) over all fractional matchings. In this paper, we study the connections between the fractional matching number and the Laplacian spectral radius of a graph. We also give some sufficient spectral conditions for the existence of a fractional perfect matching. [ABSTRACT FROM AUTHOR]

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph G is a cograph if and only if no induced subgraph of G has an eigenvalue in the interval. It is also shown that the multiplicity of any eigenvalue of a cograph G does not exceed the sum of multiplicities of 0 and as eigenvalues of G. We introduce a partial order on the vertex set of graphs G in terms of inclusions among the open and closed neighbourhoods of vertices, and conjecture that the multiplicity of any eigenvalue of a cograph G except for does not exceed the maximum size of an antichain with respect to that partial order. In two extreme cases (in particular for threshold graphs), the conjecture is shown to be true. Finally, we prove that bipartite -free graphs have no eigenvalue in the intervals and (0, 1 / 2). [ABSTRACT FROM AUTHOR]

For every real matrix A having as an eigenvector e, the all-ones vector, an upper bound is obtained for the absolute value of its eigenvalues except, in some cases, the one associated with e. Comparisons are made between this bound and some existing bounds for stochastic matrices as well as other types of matrices such as Laplacian matrices and Randić matrices. Other examples and results are also provided. An interesting recent article 'An eigenvalue localization theorem for stochastic matrices and its application to Randić matrices', Linear Algebra Appl. 2016;505:85-96, by A. Banerjee and R. Mehatari, is useful for this present paper. [ABSTRACT FROM AUTHOR]

We show how -ifications, namely, lower order matrix polynomials with the same eigenvalues as a given complex square matrix polynomial, can be used in combination with other recent results, namely, a generalization to matrix polynomials of the Cauchy radius and its enhancement, to produce eigenvalue bounds. In addition, we present a simplified derivation of -ifications for square matrices. We found that reasonably good bounds can be obtained by working with any companion matrix polynomial, and that these bounds were less computationally demanding for higher degree companion matrix polynomials than for such polynomials of degree one (linearizations). [ABSTRACT FROM AUTHOR]

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs. [ABSTRACT FROM AUTHOR]

In this paper, we extend some well-known results on the relationships of the eigenvalues and eigenvectors from matrices to the tensors case, including nonsingular tensors and their left (right) inverses, similar tensors, tensor products and , where B is a matrix. [ABSTRACT FROM AUTHOR]

Let and be two order m dimension n tensors. The Hadamard product is an order m dimension n tensor with entries . For nonnegative tensors and , we obtain some bounds on spectral radius of in terms of spectral radius and entries of and . We give some inequalities for the minimum eigenvalue and determinant of the Hadamard product involving M-tensors and Z-tensors, and use them to derive some criteria of M-tensors. [ABSTRACT FROM AUTHOR]

In this paper, we mainly focus on new inclusion sets for eigenvalues of a tensor. First, we propose new inclusion sets for eigenvalues of a tensor, which are sharper than some existing inclusion sets, and obtain the law of distribution of the number of eigenvalues for a tensor. Second, two new classes of tensors are introduced. Third, some bounds on the spectral radii for nonnegative tensors are given. Fourth, some checkable sufficient conditions for the positive definiteness (positive semidefiniteness) of some classes of even-order real symmetric tensors are obtained. [ABSTRACT FROM AUTHOR]

The distance Laplacian matrixof a connected graphGis defined to bewheredenotes the distance matrix ofGand diag(Tr) denotes the diagonal matrix of the vertex transmissions inG. Similarly, the distance signless Laplacian matrix ofGis defined asIn this paper, we present a large set of nonisomorphic graphs which are-cospectral. A simple construction of graphs which are nonisomorphic and |L|-cospectral, where |L| is the signless Laplacian matrix ofG,  is also presented. We finish this paper exhibiting nonisomorphic and-noncospectral graphs with same-energy. [ABSTRACT FROM AUTHOR]

Consider anmatrix polynomialand a setconsisting ofdistinct complex numbers. In this paper, a (weighted) spectral norm distance fromto the matrix polynomials whose spectra include the specified set, is defined and studied. An upper and a lower bound for this distance are obtained, and an optimal perturbation ofassociated to the upper bound is constructed. Numerical examples are given to illustrate the efficiency of the proposed bounds. [ABSTRACT FROM PUBLISHER]

Linear preserver problems concern the study of linear maps on matrices or operators preserving special properties. In all previous results, one needs to know certain information of the maps on an infinite number of matrices to characterize the structures of a class of linear preservers. We prove that if a linear map preserves the determinant or eigenvalues of a finite number of complex matrices of a given order, then it preserves the corresponding property of all complex matrices of order. These results weaken the conditions in some classical theorems on linear preserver problems. We also present an application of linear preservers in quantum information science. [ABSTRACT FROM PUBLISHER]

Letbe a locally finite meet semilattice. Let be a finite subset of, and letbe a complex-valued function on. The-dimensional hypermatrix of order,, given by is called the ordermeet hypermatrix onwith respect to. We consider-dimensional meet hypermatrices of order. As an example, we consider GCD hypermatrices. We examine the structure of ordermeet hypermatrices with respect to, and provide a structure theoretical result that is a generalization of a known result formeet matrices. We also give a region in which all the eigenvalues of an-dimensional ordermeet hypermatrixwith respect to a real-valuedlie, and using that we obtain results concerning positive definiteness and E-eigenvalues of meet hypermatrices. Characteristics of meet matrices and the eigenvalues of supersymmetric hypermatrices are under active research, but the eigenvalues of GCD and related hypermatrices have not hitherto been considered in the literature. [ABSTRACT FROM PUBLISHER]

The symmetric doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real-tuple to be the spectrum of ansymmetric doubly stochastic matrix. For, this problem remains open though many partial results are known. In this note, we present a new family of necessary conditions for this problem using some matrix trace inequalities. In addition, we prove that this family of new inequalities sharpen the existing known necessary conditions for the inverse spectral problem of nonnegative matrices. Finally, we prove that these necessary conditions are not sufficient for the case. [ABSTRACT FROM AUTHOR]

We give some graph theoretical formulas for the traceof a tensorwhich do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of uniform hypergraphs, we give a characterization (in terms of the traces of the adjacency tensors) of the-uniform hypergraphs whose spectra are-symmetric, thus give an answer to a question raised in Cooper and Dutle [Linear Algebra Appl. 2012;436:3268–3292]. We generalize the results in Cooper and Dutle [Linear Algebra Appl. 2012;436:3268–3292, Theorem 4.2] and Hu and Qi [Discrete Appl. Math. 2014;169:140–151, Proposition 3.1] about the-symmetry of the spectrum of a-uniform hypergraph, and answer a question in Hu and Qi [Discrete Appl. Math. 2014;169:140–151] about the relation between the Laplacian and signless Laplacian spectra of a-uniform hypergraph whenis odd. We also give a simplified proof of an expression forand discuss the expression for. [ABSTRACT FROM AUTHOR]

Letbe two Hermitian matrices. We propose new perturbation bounds on the differences between the eigenvalues ofandby the bounds of the eigenvector components. These results extend those of Nakatsukasa. [ABSTRACT FROM AUTHOR]

The eigenproblem for optimal-node matrices in max-plus algebra is shown to be solved. First, the definition of optimal-node matrix is introduced. Then for a given optimal-node matrix, it is revealed that it is easy to find all its eigenvalues and eigenvectors, which correspond to the interval time of two continuous stages and the start times of every stage of all the machines in the multi-machine interactive production process, respectively. [ABSTRACT FROM AUTHOR]

In the present paper, we deal with the perturbed operatorwhereis a closed densely defined linear operator on a separable Hilbert spaceXwith domainwhileare linear operators onXwith the same domainand satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions assuring the existence of a Riesz basis of eigenvectors of the operator. The obtained results are of importance for application to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid and to a nonself-adjoint Gribov operator in Bargmann space. [ABSTRACT FROM AUTHOR]

There are numerous min–max principles about the eigenvalues of a Hermitian matrix. The most general ones are Wielandt’s min–max principles which include the Courant–Fischer min–max principles and the trace minimization principles as special cases. In this paper, various extensions of Wielandt’s principles are obtained for a positive semi-definite pencilby which we mean thatAandBare Hermitian and there is a real numbersuch thatis positive semi-definite. [ABSTRACT FROM AUTHOR]

Two new eigenvalue inclusion regions for matrices with a constant main diagonal are given. We then apply these results to Toeplitz matrices, and obtain two regions including all eigenvalues of Toeplitz matrices. Furthermore, it is proved that the new regions are tighter than those in [Melman A. Ovals of Cassini for Toeplitz matrices,Linear and Multilinear Algebra. 2012;60:189–199]. [ABSTRACT FROM AUTHOR]

The spread of a matrix is defined as the maximum absolute value of the difference between any two eigenvalues of that matrix. Of interest, here is the spread of n × n normal matrices with entries restricted to a closed interval. In particular, we are interested in the classes of real symmetric matrices and real skew-symmetric matrices. For each of these classes, we investigate the maximum spread under certain conditions, such as small rank, and determine the structure of these matrices when their spread attains this maximum value. [ABSTRACT FROM AUTHOR]

In this article, we derive a generating function out of which various different bounds with increasing tightness for the real parts of the zeros of monic polynomials can be generated. Then we present particular bounds based on this generating function. [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]

For two square matrices that commute, we present some additive results for the Drazin inverse. We also give the application to relative perturbation of eigenvalues when the perturbed matrix commutes with the original matrix and perturbation bounds of the Drazin inverse. [ABSTRACT FROM AUTHOR]

Given a positive definite (p.d.) matrix with real entries, it is possible to construct a p.d. intraclass matrix whose diagonal and off-diagonal elements are chosen as the averages of the diagonal elements and off-diagonal elements of the former matrix. Exploiting the very special structure of the latter matrix various interesting propositions are established. Statistical applications of such matrices are surveyed. [ABSTRACT FROM AUTHOR]

Let A, X and Y be n-by-n complex matrices such that A is positive semi-definite and X, Y are contractions. We prove that if f is an increasing convex function on [0, ∞) such that f(0) ≤ 0, then the eigenvalues of f(|X*AY|) are dominated by those of X*f(A)X⊕Y* f(A)Y. Several related results are considered. [ABSTRACT FROM AUTHOR]

The structures of some important types of matrices over the quaternion skew field are discussed, and corresponding decompositions are obtained by virtue of real orthogonal matrices and real anti-symmetric matrices. In particular, the normal forms are found for several classes of quaternionic matrices. [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]

In this article we study eigenvalues and singular values of matrices. We provide some majorization inequalities for certain types of matrices. For this end we use group-induced cone orderings and normal maps. [ABSTRACT FROM AUTHOR]

Characterized are all simple undirected graphs G such that any real symmetric matrix that has graph G has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general fields), but only certain partial 2-trees guarantee maximum multiplicity 2. Among partial linear 2-trees, they are only those whose vertices can be covered by two 'parallel' induced paths. The remaining graphs that guarantee maximum multiplicity 2 are composed of certain identified families of 'exceptional' partial 2-trees that are not linear. [ABSTRACT FROM AUTHOR]

We are generally concerned with the possible lists of multiplicities for the eigenvalues of a real symmetric matrix with a given graph. Many restrictions are known, but it is often problematic to construct a matrix with desired multiplicities, even if a matrix with such multiplicities exists. Here, we develop a technique for construction using the implicit function theorem in a certain way. We show that the technique works for a large variety of trees, give examples and determine all possible multiplicities for a large class of trees for which this was not previously known. [ABSTRACT FROM AUTHOR]

According to a long standing conjecture, the geometric location of eigenvalues of doubly stochastic matrices of order n is exactly the union of regular k-gons anchored at 1 in the unit disc for 2 ≤ k ≤ n. It is easy to verify this fact for n = 2, 3. But, for n ≥ 4, it has been an open question. We show that this conjecture is wrong for n = 5. [ABSTRACT FROM AUTHOR]

The q -numerical range (0≤ q   ≤  1) of an n   ×   n matrix polynomial is defined by In this article, we investigate the boundary and the shape of Wq ( P ), using the notion of local dimension. We also obtain that the q -numerical range of first-order matrix polynomials is always simply connected. Moreover, the special cases of 2×2 matrices and matrix polynomials are considered. In particular, the boundary of the q -numerical range of a 2×2 matrix polynomial of degree m lies on an algebraic curve of degree at most 8 m . [ABSTRACT FROM AUTHOR]

The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ⩽ D and for each b, or the equation Ax + B0|x| = 0 has a nontrivial solution for some matrix B0 of a very special form, |B0| ⩽ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| λ |x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue. [ABSTRACT FROM AUTHOR]

In this article, we study a compression of normal matrices and matrix polynomials with respect to a given vector and its orthogonal complement. The numerical range of this compression satisfies special boundary properties, which are investigated in detail. The characteristic polynomial of the compression is also considered. [ABSTRACT FROM AUTHOR]