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It is shown that for positive real numbers 0 < λ 1 < ... < λ n , [ 1 β (λ i , λ j) ] , where β (⋅ , ⋅) denotes the beta function, is infinitely divisible and totally positive. For [ 1 β (i , j) ] , the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let w (n) be the n th Bell number. It is proved that [ w (i + j) ] is a totally positive matrix but is infinitely divisible only upto order 4. It is also shown that the symmetrized Stirling matrices are totally positive. [ABSTRACT FROM AUTHOR]

We consider an m × m real symmetric matrix M a (x) with a 1 , ... , a m on the main diagonal and x in all off-diagonal positions, where m ≥ 2 and a = (a 1 , ... , a m) is a given m -tuple of positive real numbers. We study the extremal problem of finding the minimum and maximum of x where M a (x) is positive semidefinite. We show that the polynomial det ⁡ M a (x) in variable x has only real roots with a unique negative root and that M a (x) is positive semidefinite (resp. definite) if and only if x lies in the closed (resp. open) interval determined by the negative and smallest positive roots. It is further shown that the negative and smallest positive root maps over m -tuples of positive real numbers contract the Thompson metric and induce new multivariate means of positive real numbers satisfying the monotonicity, homogeneity, joint concavity and super-multiplicativity. In particular, the smallest positive root map extends to such a mean of infinite variable of positive real numbers that realizes the limits for decreasing sequences, and it eventually gives rise to a shift invariant mean of bounded sequences. [ABSTRACT FROM AUTHOR]

We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M , we keep the positions of specified and unspecified entries fixed, and study the minimal completion rank. If the entries of the matrix are complex and the known entries are chosen randomly according to a continuous distribution, then for a fixed pattern of locations of specified and unspecified entries, there is a unique minimum completion rank which occurs with probability one. We call this rank the generic completion rank. Over the real numbers there can be multiple ranks that occur with positive probability; we call them typical completion ranks. We introduce these notions formally, and provide a number of inequalities and exact results on typical and generic ranks for different families of patterns of known and unknown entries. [ABSTRACT FROM AUTHOR]

Abstract The concept of an algebraically positive matrix was introduced by Kirkland, Qiao, and Zhan in 2016. A real square matrix A is said to be algebraically positive if there exists a real polynomial f such that f (A) is a positive matrix. In this paper, we give a new characterization of algebraically positive matrices, and characterize all tree sign pattern matrices that allow algebraic positivity, and all star and path sign pattern matrices that require algebraic positivity. Also, all tree sign pattern matrices of order less than 6 requiring algebraic positivity are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we propose and discuss a class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth. For an odd 2 p + 1 with a positive integer p , the problem is to construct an n × n real symmetric banded matrix with bandwidth 2 p + 1 whose m × m leading principal submatrix is a given m × m real symmetric banded matrix with bandwidth 2 p + 1 and spectrum is a given set of real numbers { λ i } i = 1 n , where the number of distinct real numbers of { λ i } i = 1 n is 2 k when m = p k , or 2 k + 1 when p k < m < p ( k + 1 ) , where m , n and k are positive integers and m < n . We point out that the well-known double dimensional (DD) problem is a special case of our proposed inverse eigenvalue problems. The necessary and sufficient condition for the solvability of the above inverse eigenvalue problem is derived, and the target real symmetric banded matrix can be constructed by the block Lanczos algorithms when the inverse eigenvalue problem is solvable. Several numerical examples show that our algorithms are feasible. Some concluding remarks are introduced. [ABSTRACT FROM AUTHOR]

Let λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥ λ 5 ≥ − λ 1 be real numbers such that ∑ i = 1 5 λ i = 0 . In [16] , O. Spector proved that a necessary and sufficient condition for λ 1 , λ 2 , λ 3 , λ 4 , λ 5 to be the eigenvalues of a traceless symmetric nonnegative 5 × 5 matrix is “ λ 2 + λ 5 ≤ 0 and ∑ i = 1 5 λ i 3 ≥ 0 ”. In this article, we show that this condition is also a necessary and sufficient condition for λ 1 , λ 2 , λ 3 , λ 4 , λ 5 to be the spectrum of a traceless bisymmetric nonnegative 5 × 5 matrix. [ABSTRACT FROM AUTHOR]

The irregularity of a simple graph G = ( V , E ) is defined as irr ( G ) = ∑ u v ∈ E ( G ) | d G ( u ) − d G ( v ) | , where d G ( u ) denotes the degree of a vertex u  ∈  V ( G ). This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. Recently, it also gains interest in Chemical Graph Theory, where it is named the third Zagreb index. In this paper, by means of the Laplacian eigenvalues and the normalized Laplacian eigenvalues of G , we establish some new spectral upper bounds for irr( G ). We then compare these new bounds with a known bound by Goldberg, and it turns out that our bounds are better than the Goldberg bound in most cases. We also present two spectral lower bounds on irr( G ). [ABSTRACT FROM AUTHOR]

In this paper we establish some necessary and sufficient solvability conditions for a system of quaternary coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. An expression of the general solution to this system is given when it is solvable. Also, a numerical example is presented to illustrate the main result of this paper. The findings of this paper widely generalize the known results in the literature. The main results are also valid over the real number field and the complex number field. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to determine all matrix representations of the real numbers. It is shown that every such representation is completely reducible, while all non-trivial irreducible representations must be of 2-dimensional and can be expressed in a unique form. It is found that those representations are essentially determined by the ways of embedding the real numbers into the complex numbers. This results in a one-to-one correspondence between the equivalent classes of irreducible representations and the equivalent classes of homomorphisms from the real number field to the complex number field. The matrix representations of the complex numbers are also determined. [ABSTRACT FROM AUTHOR]

This paper deals with nonoscillation problem about the non-autonomous linear difference system x n = A n x n − 1 , n = 1 , 2 , … , where A n is a 2 × 2 variable matrix that is nonsingular for n ∈ N . In the special case that A is a constant matrix, it is well-known that all non-trivial solutions are nonoscillatory if and only if all eigenvalues of A are positive real numbers; namely, det A > 0 , tr A > 0 and det A / ( tr A ) 2 ≤ 1 / 4 . The well-known result can be said to be an analogy of ordinary differential equations. The results obtained in this paper extend this analogy result. In other words, this paper clarifies the distinction between difference equations and ordinary differential equations. Our results are explained with some specific examples. In addition, figures are attached to facilitate understanding of those examples. [ABSTRACT FROM AUTHOR]

Positivity properties of the Hadamard powers of the matrix [ 1 + x i x j ] for distinct positive real numbers x 1 , … , x n and the matrix [ | cos ⁡ ( ( i − j ) π / n ) | ] are studied. In particular, it is shown that the n × n matrix [ ( 1 + x i x j ) r ] is positive semidefinite if and only if r is a nonnegative integer or r > n − 2 , and for every odd integer n ≥ 3 the n × n matrix [ | cos ⁡ ( ( i − j ) π / n ) | r ] is positive semidefinite if and only if r is a nonnegative even integer or r > n − 3 . [ABSTRACT FROM AUTHOR]

Let T be a tree with vertex set [ n ] = { 1 , 2 , … , n } . For each i ∈ [ n ] , let m i be a positive integer. An ordered pair of two adjacent vertices is called an arc. Each arc ( i , j ) of T has a weight W i , j which is an m i × m j matrix. For two vertices i , j ∈ [ n ] , let the unique directed path from i to j be P i , j = x 0 , x 1 , … , x d where d ⩾ 1 , x 0 = i and x d = j . Define the product distance from i to j to be the m i × m j matrix M i , j = W x 0 , x 1 W x 1 , x 2 ⋯ W x d − 1 , x d . Let N = ∑ i = 1 n m i . The N × N product distance matrix D of T is a partitioned matrix whose ( i , j ) -block is the matrix M i , j . We give a formula for det ⁡ ( D ) . When det ⁡ ( D ) ≠ 0 , the inverse of D is also obtained. These generalize known results for the product distance matrix when either the weights are real numbers, or m 1 = m 2 = ⋯ = m n = s and the weights W i , j = W j , i = W e for each edge e = { i , j } ∈ E ( T ) . [ABSTRACT FROM AUTHOR]

Let M n ( B ) denote the set of n × n ( 0 , 1 ) -matrices with Boolean arithmetic. The set of primitive matrices of exponent k , denoted E k , is the set of matrices such that A k has all nonzero entries and A j has zero entries for all j < k . For 3 ≤ k ≤ n , we characterize those linear operators that map E k to E k and E k − 1 to E k − 1 . We also characterize those linear operators that strongly preserve E k for 3 ≤ k ≤ n , that is, that map E k to E k and the complement of E k to the complement of E k . [ABSTRACT FROM AUTHOR]

The Ducci map is defined by taking a vector [ v 1 , … , v n ] T ∈ R n to [ | v 1 − v 2 | , … , | v n − v 1 | ] T . We concern ourselves with the Ducci map's action in R 3 , establishing a connection between the sequences of matrices associated with the action of the Ducci map, continued fraction representations of the real numbers, and the Stern–Brocot tree. It is shown that the real numbers have a representation via sequences of Ducci matrices, and in this Ducci number system there are essentially three types of matrix sequences: one type corresponding to the rationals, one to the quadratic irrationals, and then those types that correspond to all other real numbers. This mirrors the situation for continued fraction representations of the reals. It follows that the Ducci map on R 3 is closely connected to the Euclidean algorithm and, through its action, locates best rational approximations to the irrationals. [ABSTRACT FROM AUTHOR]

A real matrix A = ( a i j ) 1 ≤ i , j , ≤ n is said to be almost strictly totally negative if it is almost strictly sign regular with signature ε = ( − 1 , − 1 , … , − 1 ) , which is equivalent to the property that all its nontrivial minors are negative. In this paper an algorithmic characterization of nonsingular almost strictly totally negative matrices is presented. [ABSTRACT FROM AUTHOR]

Abstract: Let be one of the -dimensional classical spaces and P be a fixed subspace of type ϑ of . Let be the set of all m-dimensional totally isotropic subspaces Q of satisfying . In this paper, we compute the size of when P is non-isotropic or totally isotropic. As applications, we give lower bounds of ranks of incidence matrices of totally isotropic subspaces of over the real number field , and compute eigenvalues of some graphs. [Copyright &y& Elsevier]

Abstract: Quadratic ordering of rectangular real matrices implies Chebyshev’s inequality: If are real matrices and are real matrices such that, for all with , elementwise , then for any real , elementwise . Further, linear ordering of rectangular real matrices implies Grüss’s inequality: If, elementwise, and elementwise then elementwise . The bounds are sharp. These inequalities lead to inequalities for the spectral radius of nonnegative matrices. Linear ordering and quadratic ordering are equivalent for real scalars but not for real matrices. [Copyright &y& Elsevier]

Abstract: The celebrated Heinz inequality asserts that for , , every unitarily invariant norm and . In this paper, we present several improvement of the Heinz inequality by using the convexity of the function , some integration techniques and various refinements of the Hermite–Hadamard inequality. In the setting of matrices we prove that for real numbers α, β. [Copyright &y& Elsevier]

Abstract: In this paper, we analyse a special generalized inverse eigenvalue problem for the pair of matrices each of whose graph is a path on n vertices, by investigating the leading principal minors of the matrix . From the given data consisting of , a matrix ( ) whose graph is a path on k vertices, two column vectors and distinct real numbers λ and μ we construct and two column vectors such that is the leading principal sub-matrix of and , are the eigenpairs of , where and . Further, numerical examples are also given to demonstrate the applicability of the results developed here. [Copyright &y& Elsevier]

Abstract: Let be a positive semidefinite block matrix with square matrices M and N of the same order and denote . The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1, If, in addition, K is Hermitian, then for any real number , while if K is skew-Hermitian, then for any real number , where O is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz. [Copyright &y& Elsevier]

Abstract: It is a well-known fact that a weighting pattern matrix has a periodic realization if and only if the matrix is separable and periodic. This fact, however, does not cope with a reasonable question of period-specific realization problem: for a given weighting pattern matrix and a given positive real number, find a periodic realization corresponding to the given matrix and with a period equal to the given number. This paper answers the question by showing that the period-specific realization problem has a solution if and only if the given weighting pattern matrix is separable and has a period equal to the given positive real number. To this end, two types of period-specific realizations are constructed. One is with a constant -matrix whose dimension is not necessarily minimal among all possible period-specific realizations. The other is with a non-constant and periodically time-varying -matrix whose dimension turned out minimal among all possible period-specific realizations. [Copyright &y& Elsevier]

Abstract: This paper is concerned with subject to for a Hermitian matrix pencil A-λB, where J is diagonal and (the identity matrix of apt size). The same problem was investigated earlier by Kovač-Striko and Veselić (Linear Algebra Appl. 216 (1995) 139–158) for the case in which B is assumed nonsingular. But in this paper, B is no longer assumed nonsingular, and in fact is even allowed to be a singular pencil. It is proved, among others, that the infimum is finite if and only if is a positive semi-definite pencil (in the sense that there is a real number such that is positive semi-definite). The infimum, when finite, can be expressed in terms of the finite eigenvalues of . Sufficient and necessary conditions for the attainability of the infimum are also obtained. [Copyright &y& Elsevier]

Abstract: Ray nonsingular (RNS) matrices are a generalization of sign nonsingular (SNS) matrices from the real field to complex field. The problem of how to recognize ray nonsingular matrices is still open. In this paper, we give an algorithm to recognize the ray nonsingularity for the so called “cycle chain matrices”, whose associated digraphs are of certain simple structure. Furthermore, the cycle chain matrices that are not ray nonsingular are classified into two classes according to whether or not they can be majorized by some RNS cycle chain matrices, and the case when they can is characterized. [Copyright &y& Elsevier]

Abstract: In this paper a new method for computing the action of the matrix exponential on a vector , where is a complex matrix and is a positive real number, is proposed. Our approach is based on vector valued rational approximation where the approximants are determined by the denominator polynomials whose coefficients are obtained by solving an inexpensive linear least-squares problem. No matrix multiplications or divisions but matrix-vector products are required in the whole process. A technique of scaling and recurrence enables our method to be more effective when the problem is for fixed and many values of . We also give a backward error analysis in exact arithmetic for the truncation errors to derive our new algorithm. Preliminary numerical results illustrate that the new algorithm performs well. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the eigenstructure of finite field trigonometric transforms (FFTT). This transform family includes different types of cosine and sine transforms that correspond, in the finite field case, to the well known discrete cosine and sine transforms defined over the real numbers. More specifically, we determine the eigenvalues of the FFTT matrices and study their multiplicities. We propose procedures for constructing the associated eigenvectors. Applications of the developed theory to multiuser communication systems and error-correcting codes are also suggested. [Copyright &y& Elsevier]

Abstract: Let denote a set of all elements of weighted lattice paths with weight in the xy-plane from to such that a vertical step , a horizontal step , and a diagonal step are endowed with weights , and respectively and let denote the weight of defined bywhere is the product of the weights of all its steps in . A matrix is called a lattice path matrix with weight if for a triple , and of real numbers . In this paper, we present LDU decomposition of lattice path matrices with weight and related properties for every triple , and of real numbers, and a necessary and sufficient condition in which the symmetric lattice path matrices are positive definite. We also investigate the relationship between the lattice path matrices and generalized Pascal matrices. [Copyright &y& Elsevier]

Abstract: For a graph and a real number , the graph invariant is the sum of the th power of the non-zero Laplacian eigenvalues of . This note presents some bounds for in terms of the vertex degrees of , and a relation between and the first general Zagreb index, which is a useful topological index and has important applications in chemistry. [ABSTRACT FROM AUTHOR]

Abstract: The problem of determining necessary and sufficient conditions for a set of real numbers to be the eigenvalues of a symmetric nonnegative matrix is called the symmetric nonnegative inverse eigenvalue problem (SNIEP). In this paper we solve SNIEP in the case of trace zero symmetric nonnegative matrices. [Copyright &y& Elsevier]

Abstract: In this paper, we first give the existence of and the general expression for the solution to an inverse eigenproblem defined as follows: given a set of real -vectors and a set of real numbers , and an -by- real generalized reflexive matrix (or generalized anti-reflexive matrix ) such that and are the eigenvectors and eigenvalues of (or ), respectively, we solve the best approximation problem for the inverse eigenproblem. That is, given an arbitrary real -by- matrix , we find a matrix which is the solution to the inverse eigenproblem such that the distance between and is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm for the best approximation problem over generalized reflexive (or generalized anti-reflexive) matrices. Two numerical examples are also presented to show that our method is effective. [ABSTRACT FROM AUTHOR]

Abstract: Hermitian matrices can be thought of as generalizations of real numbers. Many matrix inequalities, especially for Hermitian matrices, are derived from their scalar counterparts. In this paper, the Hardy–Littlewood–Pólya rearrangement inequality is extended to Hermitian matrices with respect to determinant, trace, Kronecker product, and Hadamard product. [ABSTRACT FROM AUTHOR]

Abstract: For a given graph , denote by its adjacency matrix and . We say that there exist a perfect state transfer (PST) in if , for some vertices and a positive real number . Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant graphs (circulant graphs with integer eigenvalues). Some results on this topic have already been obtained by Saxena et al. (2007) , Bašić et al. (2009)  and Basić & Petković (2009) . In this paper, we show that there exists an integral circulant graph with vertices having a perfect state transfer if and only if . Several classes of integral circulant graphs have been found that have a perfect state transfer for the values of divisible by . Moreover, we prove the nonexistence of a PST for several other classes of integral circulant graphs whose order is divisible by . These classes cover the class of graphs where the divisor set contains exactly two elements. The obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer. [ABSTRACT FROM AUTHOR]

Abstract: Recently, Psarrakos obtained a lower bound, which depends on a real parameter , for the distance between the pseudospectrum of a matrix polynomial and a given point that lies out of it. In this note, we study the best choice of the parameter. As a consequence, the best lower bound among those given by Psarrakos is obtained. [Copyright &y& Elsevier]

Abstract: Let be an integer and be an edge-weighted tree. A cut on an edge of is a splitting of the edge at some point on it. A -edge-partition of is a set of subtrees induced by cuts. Given and , the max–min continuous tree edge-partition problem is to find a -edge-partition that maximizes the length of the smallest subtree; and the min–max continuous tree edge-partition problem is to find a -edge-partition that minimizes the length of the largest subtree. In this paper, -time algorithms are proposed for these two problems, improving the previous upper bounds by a factor of log (). Along the way, we solve a problem, named the ratio search problem. Given a positive integer , a (non-ordered) set of non-negative real numbers, a real valued non-increasing function , and a real number , the problem is to find the largest number in such that . We give an -time algorithm for this problem, where is the time required to evaluate the function value for any real number . [Copyright &y& Elsevier]

Abstract: Thirty years ago, G. N. de Oliveira has proposed the following completion problems: Describe the possible characteristic polynomials of , where and are square submatrices, when some of the blocks are fixed and the others vary. Several of these problems remain unsolved. This paper gives the solution, over the field of real numbers, of Oliveira’s problem where the blocks are fixed and the others vary. [Copyright &y& Elsevier]

Abstract: We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form , where is a nonnegative real number and is a th root of unity, where is the period of the matrix, which is computed from the distance between the bands. We also present a problem in the asymptotics of spectra in which such double band matrices are perturbed by banded matrices. [Copyright &y& Elsevier]

Abstract: Tensor type data are becoming important recently in various application fields. We determine the rank of a 3-tensor with 2 slices in comparison with its Kronecker canonical form over the complex and real number field. [Copyright &y& Elsevier]

Abstract: The Sylvester–Kac matrix is a tridiagonal matrix with integer entries and integer eigenvalues that appears in a variety of applicative problems. We show that it belongs to a four dimensional linear space of tridiagonal matrices that can be simultaneously reduced to triangular form. We name this space after the matrix. [Copyright &y& Elsevier]

Abstract: The real nonnegative inverse eigenvalue problem (RNIEP) is the problem of determining necessary and sufficient conditions for a list of real numbers Λ to be the spectrum of an entrywise nonnegative matrix. A number of sufficient conditions for the existence of such a matrix are known. In this paper, in order to construct a map of sufficient conditions, we compare these conditions and establish inclusion relations or independency relations between them. [Copyright &y& Elsevier]

Abstract: A capital letter means matrix. Very recently, Fujii et al. show: For every and holds for any . In fact, (★) yields Bebiano–Lemos–Providência inequality for . As an extension of (★), we show the following result. The following (i) and (ii) hold and they are equivalent: [(i)] For every , , and each , and any real number holds for and , where and . [(ii)] . If with , then for and for and . (i) implies (★). In fact, put , , in (i), then holds, and finally, replace B by , A by , by and also replace q by , then we have the desired (★). [Copyright &y& Elsevier]

Abstract: A merging function synthesizes a vector of numbers (representing any form of quantitative evaluation) into a single number (representing a consensus or collective evaluation). Assuming that all the involved numbers are drawn from a discrete set, it is shown that projection functions are the only merging functions satisfying three properties satisfied by the arithmetic mean when defined for real numbers. Another projection result is obtained under alternative assumptions when merging functions are assumed to transform matrices of numbers from a discrete set into a vector of numbers from the discrete set. [Copyright &y& Elsevier]

Abstract: Let J be the all-ones matrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ − A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ − A for exactly one value of y. We call such graphs -cospectral. It follows that is a rational number, and we prove existence of a pair of -cospectral graphs for every rational . In addition, we generate by computer all -cospectral pairs on at most nine vertices. Recently, Chesnokov and the second author constructed pairs of -cospectral graphs for all rational , where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of , and by computer we find all such pairs of -cospectral graphs on at most eleven vertices. [Copyright &y& Elsevier]

Abstract: The main result of this paper is the following: if A =(a ij ) is an inverse M-matrix, denotes the rth Hadamard power of A, then A (r) is again an inverse M-matrix for any real number r >1. This settles a conjecture proposed by Wang et al. [B.Y. Wang, X.P. Zhang, F.Z. Zhang, On the Hadamard product of inverse M-matrices, Linear Algebra Appl. 305 (2000) 23–31] affirmatively. Naturally, it shows that the conjecture of Neumann [M. Neumann, A conjecture concerning the Hadamard product of inverses of M-matrices, Linear Algebra Appl. 285 (1998) 277–290] is also valid. [Copyright &y& Elsevier]

Abstract: For every rational number x ∈(0,1), we construct a pair of graphs, one regular and one nonregular with adjacency matrices A 1 and A2, having the property that A 1 − xJ and A 2 − xJ have the same spectrum (J is the all-ones matrix). This solves a problem of Van Dam and the second author. For some values of x, we have generated the smallest examples (with respect to the number of vertices) by computer. [Copyright &y& Elsevier]

Abstract: In this paper, a class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices are essentially decomposed into the same kind subproblems for real antisymmetric matrices with smaller dimensions. We present the general solution of the constrained inverse eigenproblem for skew symmetric and centrosymmetric matrices in real number field. In addition, in corresponding solution set of the constrained inverse eigenproblem for skew symmetric and centrosymmetric matrices, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm is obtained. [Copyright &y& Elsevier]

Abstract: Let S n be the positive real symmetric matrix of order n with (i, j) entry equal to , and let x be a positive real number. Eigenvalues of the Hadamard (or entry wise) power are considered. In particular for k a positive integer, it is shown that both the Perron root and the trace of are approximately equal to . [Copyright &y& Elsevier]

Abstract: A ray pattern is a complex matrix each of whose entries is either 0 or a ray eiθ , where θ is a real number. For a ray pattern A =[a st ], we define the ray pattern of A, where if a st ≠0 and if a st =0. In this paper, we first show that an irreducible powerful ray pattern A is ray diagonally similar to ω∣A∣ for some ray ω. By using this representation, we obtain several results on irreducible powerful ray patterns and irreducible periodic ray patterns. Then we show that the number of such rays ω is k(A), where k(A) is the index of imprimitivity of A. As an application to complex matrices, we generalize the Perron–Frobenius Theorem to a subclass of complex matrices. [Copyright &y& Elsevier]

Abstract: Let G =(V, E) be a simple graph and {λ 1(G),…, λ n (G)} be its adjacency spectrum. It is easy to see that if an edge is added between two isolated vertices, then one zero eigenvalue increases by 1, and another zero eigenvalue decreases by 1. Let G + be a connected graph obtained from G by adding an edge e ∉ E(G). In this paper, it will be proved that the spectrum of G + is different from that of G only in two places with one eigenvalue increases by m and another eigenvalue decreases by m, where m >0 is a rational number, if and only if G is an empty graph with order 2. It will also be proved that one cannot construct a new adjacency integral connected graph with order n ⩾3 from a known one by adding an edge. [Copyright &y& Elsevier]

Abstract: An n×n matrix over the field of real numbers is a doubly negative matrix if it is symmetric, negative definite and entry-wise negative. In this paper, we are interested in the doubly negative matrix completion problem, that is when does a partial matrix have a doubly negative matrix completion. In general, we cannot guarantee the existence of such a completion. In this paper, we prove that every partial doubly negative matrix whose associated graph is a p-chordal graph G has a doubly negative matrix completion if and only if p=1. Furthermore, the question of completability of partial doubly negative matrices whose associated graphs are cycles is addressed. [Copyright &y& Elsevier]

Let S be a subset of the set of real numbers R. A is called S-factorizable if it can be factorized as A=BBT with bij∈S. The smallest possible number of columns of B in such factorization is called the S-rank of A and is denoted by rankSA. If S is a set of nonnegative numbers, then A is called S-cp. The aim of this work is to study {0,1}-cp matrices. We characterize {0,1}-cp matrices of order less than 4, and give a necessary and sufficient condition for a matrix of order 4 with some zero entries, to be {0,1}-cp. We show that a nonnegative integral Jacobi matrix is {0,1}-cp if and only if it is diagonally dominant, and obtain a necessary condition for a 2-banded symmetric nonnegative integral matrix to be {0,1}-cp. We give formulae for the exact value of the {0,1}-rank of integral symmetric nonnegative diagonally dominant matrices and some other {0,1}-cp matrices. [Copyright &y& Elsevier]

Abstract: Let f be a function with a regular strictly positive function and a real number α in . In a previous paper for such a number α we have obtained the asymptotic behaviour of the entries of the inverse of the Toeplitz matrix when N goes to infinity. These results allow us to give trace formulas, which extend a classical expression of Szegö''s limits theorems. [Copyright &y& Elsevier]