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This paper considers the usual compound matrix, the additive compound matrix, and the discriminant matrix. An investigation of the Jordan structure of discriminant matrices is initiated. The primary objective of the paper is the determination of the Jordan form of the discriminant of one Jordan block. This Jordan form has a very simple pattern. A number of related results are obtained. The main results are facilitated by the new precedence relation ordering and the block-persymmetric property of the matrices. Examples of the discriminant matrices and Jordan forms are provided. [ABSTRACT FROM AUTHOR]

This note supplies additions to our paper quoted in the title on integral spans of tight frames in Euclidean spaces. In that previous paper, we considered the case of an equiangular tight frame (ETF), proving that if its integral span is a lattice then the frame must be rational, but overlooking a simple argument in the reverse direction. Thus our first result here is that the integral span of an ETF is a lattice if and only if the frame is rational. Further, we discuss conditions under which such lattices are eutactic and perfect and, consequently, are local maxima of the packing density function in the dimension of their span. In particular, the unit ( 276 , 23 ) equiangular tight frame is shown to be eutactic and perfect. More general tight frames and their norm-forms are considered as well, and definitive results are obtained in dimensions two and three. [ABSTRACT FROM AUTHOR]

A graph G is said to be determined by its generalized spectrum (DGS for short) if whenever Γ is a graph such that Γ and G are cospectral with cospectral complements, then Γ is isomorphic to G . Let G ∪ H be the disjoint union of graphs G and H . In this paper, we give a simple sufficient condition, under which we show that G ∪ H is DGS if and only if both G and H are DGS. In particular, let H = { x } be a singleton graph, we show that if gcd ⁡ ( a n , det ⁡ ( W ( G ) ) ) = 1 and a n is square-free, then G ∪ { x } is DGS if and only if G is DGS, where a n is the constant term of the characteristic polynomial of G and W ( G ) is the walk-matrix of G . It is noticed that in Wang and Xu [9] , the authors gave a sufficient condition for G ∪ { x } to be DGS if G is DGS. However, they missed the condition that a n is square-free in their theorem, and the result obtained is incorrect. We found a counterexample to their result without this condition and give a correct version of the result accordingly in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new decomposition (called the core-EP decomposition in the present paper) for square matrices and give some of its applications. By applying the decomposition, we derive several characterizations of the core-EP inverse, introduce two orders – the core-EP order and the core-minus partial order, and characterize their properties. We also give some properties of the decomposition. [ABSTRACT FROM AUTHOR]

This paper explores M ( R ∞ 4 ) , the group of orientation preserving Möbius transformations acting in R ∞ 4 . On the one hand M ( R ∞ 4 ) is given by the group of 2 × 2 matrices over the quaternions H with determinant D derived from the corresponding 4 × 4 matrices over the complex numbers C . On the other hand we know that, in general, M ( R ∞ n ) may be given in terms of 2 × 2 matrices over the Clifford algebra C n with n − 1 generators. Thus when n = 4 , M ( R ∞ 4 ) is given in terms of 2 × 2 matrices with entries drawn from C 4 and determinant Δ defined in terms of the entries of the given matrix. We note that the skew field H may be considered as a Clifford algebra C 3 based on two generators i and j or more generally i 1 and i 2 where i j = k or i 1 i 2 = k , while the set of elements { 1 , i , j , k } form a basis of H regarded as a 4-dimensional real vector space. Thus H is embedded in C 4 . In the paper we reconcile the two representations of M ( R ∞ 4 ) by comparing the generating sets of the underlying groups of matrices. A relationship between a determinants D and Δ is also exposed. [ABSTRACT FROM AUTHOR]

This paper presents necessary and sufficient conditions for a square matrix over an arbitrary field to be expressed as a product of idempotent and square-zero matrices. Since the similarity structures of such factors are completely determined by their rank, we also investigate the ranks that these factors may have. The case where the factors are all idempotent or all square-zero have been investigated by various authors, some of whom are listed in the references of this paper. [ABSTRACT FROM AUTHOR]

The paper is devoted to the structure and the asymptotics of the eigenvector matrix of Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The results extend existing results on banded Toeplitz matrices to full Toeplitz matrices with temperate decay of the entries in the first row and column. We establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors. These formulas are in terms of the Wiener–Hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue. The paper does not aim at providing numerical algorithms. Its main purpose is rather to reveal the structure underneath the eigenvectors, which are shown to be the sum of two harmonics and certain edge corrections. [ABSTRACT FROM AUTHOR]

This paper presents some analytic methods for studying uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues λ and λ min of 2-graphs. First, these two parameters are extended to weighted uniform hypergraphs; second, the eigenvalues-numbers λ and λ min are extended to eigenvalues-functions λ (p) and λ min (p), which also encompass other graph parameters like the Lagrangian and the number of edges. In this way the functions λ (p) and λ min (p) seamlessly join spectral and traditional results in hypergraphs. In particular, this new viewpoint helps to show that spectral extremal and edge extremal problems are asymptotically equivalent. Naturally, all results about λ (p) and λ min (p) also extend spectral hypergraph theory, but delve into deeper problems than before. In fact, the resulting theory is new even for 2-graphs, where some well-settled topics become research challenges again. The paper covers a multitude of topics, with more than a hundred concrete statements to underpin an analytic theory for hypergraphs. Essential among these topics are a Perron–Frobenius type theory and methods for extremal hypergraph problems. Many open problems are raised and directions for possible further research are outlined. [ABSTRACT FROM AUTHOR]

Abstract: Richard Brualdi proposed in Stevanivić (2007) [6] the following problem: (Problem AWGS.4) Let and be two nonisomorphic graphs on n vertices with spectra respectively. Define the distance between the spectra of and as Define the cospectrality of by Let In this paper we study Problem A and determine the cospectrality of certain graphs by the Euclidian distance. Let denote the complete graph on n vertices, denote the null graph on n vertices and denote the disjoint union of the with isolated vertices, where . In this paper we find , , ( ) and . [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, we apply a complete pivoting strategy to compute the left-looking version of AINV preconditioner for linear systems. As the preprocessing, the MultiLevel Nested Dissection reordering has also been applied. We have used this preconditioner as the right preconditioner for several linear systems where the coefficient matrices have been downloaded from the University of Florida Sparse Matrix Collection. Numerical experiments presented in this paper indicate the effectiveness of such a complete pivoting on the quality of left-looking version of AINV preconditioner. [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: This paper studies Symmetric Determinantal Representations (SDR) in characteristic 2, that is the representation of a multivariate polynomial P by a symmetric matrix M such that , and where each entry of M is either a constant or a variable. We first give some sufficient conditions for a polynomial to have an SDR. We then give a non-trivial necessary condition, which implies that some polynomials have no SDR, answering a question of Grenet et al. A large part of the paper is then devoted to the case of multilinear polynomials. We prove that the existence of an SDR for a multilinear polynomial is equivalent to the existence of a factorization of the polynomial in certain quotient rings. We develop some algorithms to test the factorizability in these rings and use them to find SDRs when they exist. Altogether, this gives us polynomial-time algorithms to factorize the polynomials in the quotient rings and to build SDRs. We conclude by describing the case of Alternating Determinantal Representations in any characteristic. [Copyright &y& Elsevier]

A classical result due to Eneström and Kakeya gives some bounds for the moduli of the zeros of polynomials having a monotone sequence of non-negative (real) coefficients. The main subject of the present paper is a study of this fact with a view to the recurrence relations fulfilled by systems of orthogonal polynomials on the unit circle. In particular, we will be interested in the special case, where the zeros of the polynomials in question are not located on the boundary of the estimate which occurs in the Eneström–Kakeya theorem. Among other things, we will give characterizations of this case in terms of orthogonal polynomials and present an alternative approach to a well-known characterization due to Hurwitz. Furthermore, we will give some insight how one can apply the main results of this paper in the context of positive Hermitian Toeplitz matrices. [ABSTRACT FROM AUTHOR]

Abstract: Let A be the adjacency matrix of a graph X and suppose . We view A as acting on and take the standard basis of this space to be the vectors for u in . hysicists say that we have perfect state transfer from vertex u to v at time if there is a scalar such that (Since is unitary, .) For example, if X is the d-cube and u and v are at distance d then we have perfect state transfer from u to v at time . Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from u to v if there is a complex number and, for each positive real there is a time t such that Again we necessarily have . In a recent paper Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path if and only is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for and .) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we extend the theory of pretty good state transfer. We provide what is only the second family of graphs where pretty good state transfer occurs. The graphs we use are the double-star graphs , these are trees with a vertex of degree adjacent to a vertex of degree , and all other vertices of degree one. We prove that perfect state transfer does not occur in any graph in this family. We show that if , then there is pretty good state transfer in between the two end vertices adjacent to the vertex of degree three. If , we prove that there is never perfect state transfer between the two vertices of degree at least three, and we show that there is pretty good state transfer between them if and only these vertices both have degree and is not a perfect square. Thus we find again the the existence of perfect state transfer depends on a number theoretic condition. It is also interesting that although no double stars have perfect state transfer, there are some that admit pretty good state transfer. [Copyright &y& Elsevier]

Abstract: In a recent paper [D. Zhang, Z. Lin, Y. Liu, On eigenvalues and equivalent transformation of trigonometric matrices, Linear Algebra Appl. 436 (2012) 71–78], the authors motivated and discussed a trigonometric matrix that arises in the design of finite impulse response (FIR) digital filters. The eigenvalues of this matrix shed light on the FIR filter design, so obtaining them in closed form was investigated. Zhang et al. proved that their matrix is rank-4 and they conjectured closed form expressions for its eigenvalues. This paper studies trigonometric matrices more general than theirs, deduces their rank, and derives closed-forms for their eigenvalues. As a corollary, it yields an elementary proof of the conjecture in the aforementioned paper. [Copyright &y& Elsevier]

Abstract: This paper improves on generalized properties of a family of iterative methods to compute the approximate inverses of square matrices originally proposed in [1]. And while the methods of [1] can be used to compute the inner inverses of any matrix, it has not been proved that these sequences converge (in norm) to a fixed inner inverse of the matrix. In this paper, it is proved that the sequences indeed are convergent to a fixed inner inverse of the matrix which is the Moore–Penrose inverse of the matrix. The convergence proof of these sequences is given by fundamental matrix calculus, and numerical experiments show that the third-order iterations are as good as the second-order iterations. [Copyright &y& Elsevier]

Abstract: In this paper, a symmetric nonnegative matrix with zero diagonal and given spectrum, where exactly one of the eigenvalues is positive, is constructed. This solves the symmetric nonnegative eigenvalue problem (SNIEP) for such a spectrum. The construction is based on the idea from the paper Hayden, Reams, Wells, “Methods for constructing distance matrices and the inverse eigenvalue problem”. Some results of this paper are enhanced. The construction is applied for the solution of the inverse eigenvalue problem for Euclidean distance matrices, under some assumptions on the eigenvalues. [Copyright &y& Elsevier]

Abstract: In this paper, we will derive the condensed Cramer’s rule of Werner for minimal-norm least-squares solution of linear equations from the Cramer’s rule of Ben-Israel and Verghese. In addition, a new condensed Cramer’s rule will be obtained in this paper. [Copyright &y& Elsevier]

Abstract: This paper studies the Laplacian spectral characterization of some graph products. We consider a class of connected graphs: , and characterize all graphs such that the products are L-DS graphs. The main result of this paper states that, if , except for , is a L-DS graph, so is the product . In addition, the L-cospectral graphs with have been found. [Copyright &y& Elsevier]

Abstract: Propositions 24 and 25 of Book I of Euclid’s Elements state the fairly obvious fact that if an angle in a triangle is increased without changing the lengths of its arms, then the length of the opposite side increases, and conversely. A satisfactory analogue that holds for orthocentric tetrahedra is established by S. Abu-Saymeh, M. Hajja, M. Hayajneh in a yet unpublished paper, where it is also shown that no reasonable analogue holds for general tetrahedra. In this paper, the result is shown to hold for orthocentric d-simplices for all . The ingredients of the proof consist in finding a suitable parametrization (by a single real number) of the family of orthocentric d-simplices whose edges emanating from a certain vertex have fixed lengths, and in making use of properties of certain polynomials and of Gram and positive definite matrices and their determinants. [Copyright &y& Elsevier]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: what is the structure of a connected graph G with rank 5? or equivalently, what is the structure of a connected n-vertex graph G whose adjacency matrix has nullity ? In this paper, we completely characterize connected graphs G whose adjacency matrix has rank 5. [Copyright &y& Elsevier]

Abstract: It is well known that if T is invertible and is its inverse, then is the inverse of for . The main purpose of this paper is to study the following problem: if is an inner inverse of T and , is an inner inverse of ? If the answer is no, when is an inner inverse of ? In this paper, we give a complete answer to these problems and provide some characterizations for to be an inner inverse of in Banach spaces. Utilizing these results, we investigate similar problems for the generalized inverse, {1,3}-inverse, {1,4}-inverse, {1,5}-inverse, {1,2,3}-inverse, {1,2,4}-inverse, Moore–Penrose inverse, group inverse and Drazin inverse. The results obtained in this paper extend and improve many recent results in this area. [Copyright &y& Elsevier]

Abstract: The paper is devoted to the eigenvectors of Hessenberg Toeplitz matrices whose symbol has a power singularity. We describe the structure of the eigenvectors and prove an asymptotic formula which can be used to compute individual eigenvectors effectively. The symbols of our matrices are special Fisher–Hartwig symbols, and the theorem of this paper confirms and makes more precise a conjecture by Dai, Geary, and Kadanoff of 2009 in a particular case. [Copyright &y& Elsevier]

Abstract: In this paper, we slightly generalize the notion of G-matrices, which has been recently introduced. A real nonsingular matrix A is called a G-matrix if there exist nonsingular diagonal matrices and such that . We generalize this definition to the case where A can be singular. We say that a real matrix A, which is not necessarily square, is a generalized G-matrix (GG-matrix) if there exist nonsingular diagonal matrices and such that is a g-inverse of A. The main purpose of this paper is to show that any generalized Cauchy matrix is a GG-matrix. [Copyright &y& Elsevier]

Abstract: This paper extends Hua’s theorem on the geometry of rectangular matrices over a division ring to the case of Bezout domains. Let be integers , R an be two Bezout domains. Assume that is an adjacency preserving bijective map in both directions. Further, assume that is a local ring, or is an invertibility preserving map, or is an additive map. This paper obtains the algebraic formulas of . As applications, the ring semi-isomorphisms from to are characterized, and the group isomorphisms from to are discussed. [Copyright &y& Elsevier]

Abstract: Let be a solvable Lie subalgebra of the Lie algebra ( as a vector space). Let be polynomials in the commuting variables with coefficients in . For matrices , let and letIn this paper, we prove that, for , if one value of the matrix-valued function (the value depends on the product order of the variables) is nilpotent, then, (a) all values of are nilpotent; (b) all values of (again depends on the product order of the variables) are nilpotent, and one value is 0. This generalizes the recent result in and makes his result accurate. The main tool we use in this paper is the representation theory of solvable Lie algebras. [Copyright &y& Elsevier]

Abstract: In this paper theoretical results regarding a generalized minimum rank matrix approximation problem in the spectral norm are presented. An alternative solution expression for the generalized matrix approximation problem is obtained. This alternative expression provides a simple characterization of the achievable minimum rank, which is shown to be the same as the optimal objective value of the classical problem considered by Eckart–Young–Schmidt–Mirsky, as long as the generalized problem is feasible. In addition, this paper provides a result on a constrained version of the matrix approximation problem, establishing that the later problem is solvable via singular value decomposition. [Copyright &y& Elsevier]

Abstract: In this paper we deal with two aspects of the minimum rank of a simple undirected graph G on n vertices over a finite field with q elements, which is denoted by . In the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on n vertices over is , were . In the second part of this paper we assume that G contains a clique on k-vertices. We show that if q is not a prime then for and . It is known that for , and . We show that for and each there exists a graph G such that . For , and we show that . [Copyright &y& Elsevier]

Abstract: This paper investigates the relation between the eigenvalues of a Euclidean distance matrix (EDM) and those of the corresponding positive semidefinite matrix. More precisely, let and be two EDMs and let and be the corresponding positive semidefinite matrices such that . In this paper, when the eigenvalues of majorizes those of , a condition under which the eigenvalues of majorizes those of is derived. [Copyright &y& Elsevier]

Abstract: In this short paper, we mainly aim to study the generalized ridge estimator in a linear regression model. Through matrix techniques including Hadamard product and derivative of a vector, the globally optimal generalized ridge estimator is derived under the generalized cross-validation criterion from the theoretical point of view. It will be seen that the notion of linearized ridge estimator plays an important role in the process. A numerical example is applied to illustrate the main results of the paper. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a class of regular bipartite graphs whose biadjacency matrices are circulant matrices – a generalization of circulant graphs which happen to be bipartite – and we describe some of their properties. Notably, we compute upper and lower bounds for the zero forcing number for such a graph based only on the parameters that describe its biadjacency matrix. The main results of the paper characterize the bipartite circulant graphs that achieve equality in the lower bound and compute their minimum ranks. [Copyright &y& Elsevier]

Abstract: In this paper, we study the structure and properties of those -dimensional Lie algebras associated with either summed structures of complete graphs or some families of digraphs, having into consideration that all these combinatorial structures are made up of vertices. Our main goal is to obtain criteria determining when a Lie algebra is associated with some of combinatorial structures considered in this paper, as well as to study the properties of those structures in order to use them as a tool for classifying the types of Lie algebras associated with them. [Copyright &y& Elsevier]

Abstract: The paper consists of two parts. In the first part, we consider two matrices that appear in the literature in the study of irreducible Markov chains. The first matrix N is equal to the mean first passage of the Markov chain except on the diagonal where N vanishes. The other matrix K is equal to , where J is the all-1 matrix, A is the identity minus the transition matrix of the Markov chain, and is the diagonal matrix whose diagonal entries are the corresponding diagonal entries of the group inverse of A. Both N and K are known to be invertible. We show that the diagonal entries of and of are strictly negative in sufficiently high dimensions ( for N and for K). These results lead to a number of inequalities of independent interest, one of which we study in greater detail probabilistically. In the second part of the paper, we address a problem raised by Kemeny and Snell of determining whether a given Markov chain is primitive only from its first mean passage matrix, without having to compute the transition matrix. We derive several simple conditions of the mean first passage matrix which are helpful in determining whether the corresponding transition matrix is primitive. [Copyright &y& Elsevier]

Abstract In this paper, we study the problem of estimating certain functionals generated by some maps defined on a linear space endowed with the structure of a normal decomposition system. We reduce such task to the case of a subsystem of the given system. Thus we develop a result of Drury (2004) [4] stated for functionals induced by isoclinally metaconvex maps on Hermitian matrices. Some applications are shown. [ABSTRACT FROM AUTHOR]

Abstract In this paper we present some characterizations for quasi-arithmetic operator means (among them the arithmetic and harmonic means) on the positive definite cone of the full algebra of Hilbert space operators, and also for the Kubo–Ando geometric mean on the positive definite cone of a general non-commutative C ⁎ -algebra. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we obtain an inequality for unitarily invariant norms which unifies the arithmetic–geometric mean inequality and the Hölder inequality for unitarily invariant norms. [ABSTRACT FROM AUTHOR]

Abstract In this paper the spherical quasi-convexity of quadratic functions on spherically convex sets is studied. Several conditions characterizing the spherical quasi-convexity of quadratic functions are presented. In particular, conditions implying spherical quasi-convexity of quadratic functions on the spherical positive orthant are given. Some examples are provided as applications of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we show that if X is a two-dimensional real normed space such that its unit sphere contains a line segment with the distance between its endpoints being greater than 1, then X has the Mazur–Ulam property. That is, every isometry from the unit sphere of X onto the unit sphere of any normed space Y can be extended to a linear isometry from X onto Y. [ABSTRACT FROM AUTHOR]

Abstract: Suppose is a graph and are the eigenvalues of . The Estrada index of is defined as the sum of , . In this paper some new upper bounds for the Estrada index of bipartite graphs are presented. We apply our result on a -fullerene to improve our bound given in an earlier paper. [Copyright &y& Elsevier]

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: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: What is the structure of a connected graph with rank 4? This question has not yet been fully answered in the literature, and only some partial results are known. In this paper we resolve this question by completely characterizing graphs G whose adjacency matrix has rank 4. [Copyright &y& Elsevier]

Abstract: In an earlier paper the authors examined the problem of selecting rows of a matrix so that the resulting matrix is as “non-singular” as possible. However, the proof of the key result in that paper is not constructive. In this note we give a constructive proof for that result. In addition, we examine a case where as non-singular as possible means maximizing a determinant and provide a new bound and a constructive proof for this case also. [Copyright &y& Elsevier]

Abstract: In a recent paper [7], Gowda et al. extended Ostrowski–Schneider type inertia results to certain linear transformations on Euclidean Jordan algebras. In particular, they showed that whenever by the min–max theorem of Hirzebruch, where the inertia of an element x in a Euclidean Jordan algebra is defined bywith , , and denoting, respectively, the number of positive, negative, and zero eigenvalues, counting multiplicities. In this paper, we present a Peirce decomposition version of Wimmer’s result [13] and show that it is equivalent to the above result. In addition, we extend Higham and Cheng’s result ([8], Lemma 4.2) to the setting of Euclidean Jordan algebras. [Copyright &y& Elsevier]

Abstract: Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results. Previous work has focused on characterizing -quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between -quasiseparable matrices and the subclass of -quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices. [Copyright &y& Elsevier]

Abstract: Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results. Previous work has focused on characterizing -quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between -quasiseparable matrices and the subclass of -quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices. [Copyright &y& Elsevier]

Let ( X , d ) be a finite metric space. This paper first discusses the spectrum of the p -distance matrix of a finite metric space of p -negative type and then gives upper and lower bounds for the so called gap Γ ( X , p ) of a finite metric space of strict p -negative type. Furthermore estimations for Γ ( X , p ) under a certain gluing construction for finite metric spaces are given and finally be applied to finite ultrametric spaces. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For any real α ∈ [ 0 , 1 ] , Nikiforov [8] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . In this paper, we give some results on the eigenvalues of A α ( G ) for α > 1 / 2 . In particular, we characterize the graphs with λ k ( A α ( G ) ) = α n − 1 for 2 ≤ k ≤ n . Moreover, we show that λ n ( A α ( G ) ) ≥ 2 α − 1 if G contains no isolated vertices. [ABSTRACT FROM AUTHOR]

In this paper, we give a new angle to interpret Riordan arrays by showing that every Riordan array can be expressed as a Krylov matrix. We then use this idea to obtain some groups containing the Riordan group as a subgroup. Moreover, we study Lie algebras for the extended Riordan groups as Lie groups. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce new necessary and sufficient conditions for an Euclidean distance matrix to be multispherical. The class of multispherical distance matrices studied in this paper contains not only most of the matrices studied by Hayden et al. (1996) , but also many other multispherical structures that do not satisfy the conditions in Hayden et al. (1996) . We also study the information provided by the origin of coordinates when it is placed at the center of the spheres and the origin representation property is satisfied. These vectors associated with the origin of coordinates generate a number of supporting hyperplanes for a family of multispherical matrices and also describe part of the null space of the corresponding distance matrices. [Copyright &y& Elsevier]