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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]

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: 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: 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: 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: 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: Let and be respectively the adjacency matrix and the degree matrix of a graph . The signless Laplacian matrix of is defined as . The -spectrum of is the set of the eigenvalues together with their multiplicities of . The -index of is the maximum eigenvalue of . The possibilities for developing a spectral theory of graphs based on the signless Laplacian matrices were discussed by Cvetković et al. [D. Cvetković, P. Rowlinson, S.K. Simić, Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007) 155–171]. In the latter paper the authors determine the graphs whose -index is in the interval . In this paper, we investigate some properties of -spectra of graphs, especially for the limit points of the -index. By using these results, we characterize respectively the structures of graphs whose the -index lies in the intervals , and , where . [Copyright &y& Elsevier]

Abstract: In this paper, we study potential convergence of modulus patterns. A modulus pattern is convergent if all complex matrices in (i.e. all matrices with modulus pattern ) are convergent. A modulus pattern is potentially (absolutely) convergent if there exists a (nonnegative) convergent matrix in . We also introduce types of potential convergence that correspond to diagonal and D-convergence, studied in [E. Kaszkurewicz, A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhauser, 2000]. Convergent modulus patterns have been completely characterized by Kaszkurewicz and Bhaya [E. Kaszkurewicz, A. Bhaya, Qualitative stability of discrete-time systems, Linear Algebra Appl. 117 (1989) 65–71]. This paper presents some techniques that can be used to establish potential convergence. Potential absolute convergence and potential diagonal convergence are shown to be equivalent, and their complete characterization for modulus patterns is given. Complete characterizations of all introduced types of potential convergence for modulus patterns are also presented. [Copyright &y& Elsevier]

Abstract: In this paper, we completely solve a problem whose two-interval case has been studied recently by Wang et al. in [A. Wang, J. Sun, A. Zettl, Two-interval Sturm–Liouville operators in modified Hilbert spaces, J. Math. Anal. Appl. 328 (2007) 390–399]. The problem is to give an explicit description of all boundary conditions defining the self-adjoint differential operators associated with several Sturm–Liouville equations together, i.e., the so-called multi-interval Sturm–Liouville problems. The explicit description obtained in this paper only uses hermitian matrices. As an application, we then show that a generic self-adjoint boundary condition can not come from any direct sum of self-adjoint operators associated with the individual Sturm–Liouville equations, and it depends non-trivially on the multiples used in the definition of the so-called maximum operator. [Copyright &y& Elsevier]

Abstract: In an earlier paper [J.J. Hunter, Mixing times with applications to perturbed Markov chains, Linear Algebra Appl. 417 (2006) 108–123] the author introduced the statistic as a measure of the “mixing time” or “time to stationarity” in a finite irreducible discrete time Markov chain with stationary distribution and as the mean first passage time from state i to state j of the Markov chain. This was shown to be independent of the initial state i with for all i, minimal in the case of a periodic chain, yet can be arbitrarily large in a variety of situations. In this paper we explore the variance of the mixing time , starting in state i. The are shown to depend on i and an exploration of recommended starting states, given knowledge of the transition probabilities, is considered. As a preamble, a study of the computation of second moments of the first passage times, , and the variance of the first passage times, in a discrete time Markov chain is carried out leading to some new results. [Copyright &y& Elsevier]

Abstract: The concept of a hypergeneralized projector as a matrix H satisfying H 2 = H †, where H † denotes the Moore–Penrose inverse of H, was introduced by Groß and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]. In the present paper, the problem of when a linear combination c 1 H 1 + c 2 H 2 of two hypergeneralized projectors H 1, H 2 is also a hypergeneralized projector is considered. Although, a complete solution to this problem remains unknown, this article provides characterizations of situations in which (c 1 H 1 + c 2 H 2)2 =(c 1 H 1 + c 2 H 2)† derived under certain commutativity property imposed on matrices H 1 and H 2. The results obtained substantially generalize those given in the above mentioned paper by Groß and Trenkler. [Copyright &y& Elsevier]

Abstract: Tsallis relative operator entropy was defined as a parametric extension of relative operator entropy and the generalized Shannon inequalities were shown in the previous paper. After the review of some fundamental properties of Tsallis relative operator entropy, some operator inequalities related to Tsallis relative operator entropy are shown in the present paper. Our inequalities give the upper and lower bounds of Tsallis relative operator entropy. The operator equality on Tsallis relative operator entropy is also shown by considering the tensor product. This relation generalizes the pseudoadditivity for Tsallis entropy. As a corollary of our operator equality derived from the tensor product manipulation, we show several operator inequalities including the superadditivity and the subadditivity for Tsallis relative operator entropy. Our results are generalizations of the superadditivity and the subadditivity for Tsallis entropy. [Copyright &y& Elsevier]

Abstract: This paper studies the linear complementarity problem LCP(M,q) over the second-order (Lorentz or ice-cream) cone denoted by , where M is a n×n real square matrix and q∈Rn. This problem is denoted as SOLCP(M,q). The study of second-order cone programming problems and hence an independent study of SOLCP is motivated by a number of applications. Though the second-order cone is a special case of the cone of squares (symmetric cone) in a Euclidean Jordan algebra, the geometry of its faces is much simpler and hence an independent study of LCP over may yield interesting results. In this paper we characterize the R0-property (, and 〈x,M(x)〉=0⇒x=0) of a quadratic representation Pa(x):=2a∘(a∘x)−a2∘x of Λn for a, x∈Λn where ‘∘’ is a Jordan product and show that the R0-property of Pa is equivalent to stating that SOLCP(Pa,q) has a solution for all q∈Λn. [Copyright &y& Elsevier]

It is well known that not all sign idempotent sign pattern matrices allow idempotence. In this paper, we found a necessary and sufficient condition for a sign idempotent sign pattern matrix to allow idempotence. In addition, we investigated the nonnegativity of sign idempotent sign pattern matrices that allow idempotence. Finally, we show that if A is a sign idempotent sign pattern matrix, then A allows idempotence if and only if | A | = [ | a i j | ] also allows idempotence. [ABSTRACT FROM AUTHOR]

In this paper the concept of B-majorization is studied and some facts about this relation are found. Next, the linear preservers of B-majorization on R n and M n , m will be characterized. It is also shown that the nonzero linear preservers and strong linear preservers of B-majorization on R n are the same. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, some connections of dense alternating sign matrices with total unimodularity, combined matrices, and generalized complementary basic matrices are explored. [Copyright &y& Elsevier]

Abstract: We prove a conjecture by Bergweiler and Eremenko on the traces of elements of modular group in this paper. [Copyright &y& Elsevier]

Abstract: The paper provides the construction of association scheme on the subspaces of type in singular symplectic geometry over finite fields. All intersection numbers of the scheme are computed. At last, an authentication code with perfect secrecy from the association scheme is construction. [Copyright &y& Elsevier]

Abstract: In this paper, the positive and negative inertia index of trees, unicyclic graphs and bicyclic graphs are discussed, the methods of calculating them are obtained, and an inequality about the difference between positive and negative inertia index is proved. Moreover, a conjecture is proposed. [Copyright &y& Elsevier]

Abstract: In this paper, we define the generalized Akiyama-Tanigawa matrix and investigate its relationship with the r-Stirling numbers of the second kind. Also, we give the generating function for any column or row of the generalized Akiyama-Tanigawa matrix. As a result, we find some new identities about r-Stirling numbers of the second kind. In addition, some applications are given. [Copyright &y& Elsevier]

Abstract: This paper focuses on spectral properties of self-adjoint subspaces in the product space of a Hilbert space. Several classifications of the spectrum of a self-adjoint subspace are introduced, including discrete spectrum, essential spectrum, continuous spectrum, singular continuous spectrum, absolutely continuous spectrum, and singular spectrum. Their relationships between the subspace and its operator part are established, respectively. It is shown that all self-adjoint subspace extensions of a closed Hermitian subspace have the same essential spectrum. As a simple application, it is shown that every self-adjoint subspace extension of the corresponding minimal subspace to a singular second-order symmetric linear difference equation has a bounded spectrum under some conditions. This shows that there is an essential difference between spectral properties of singular symmetric difference operators and classical regular or singular symmetric differential operators. [Copyright &y& Elsevier]

Abstract: The first author and Hall defined recently a G-matrix as a real nonsingular matrix A such that there exist diagonal matrices and for which . The class of G-matrices was shown to possess interesting properties. In this paper, some new characterizations are found and extensions to rectangular and complex matrices are discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we exhibit new and sharper upper bounds of the spread of a matrix. [Copyright &y& Elsevier]

Abstract: The Rado–Horn theorem provides necessary and sufficient conditions for when a family of vectors can be partitioned into a fixed number of linearly independent sets. Such partitions exist if and only if every subfamily of the vectors satisfies the so-called Rado–Horn inequality. In this paper we provide an elementary proof of the Rado–Horn theorem as well as results for the redundant case. Previous proofs give no information about how to actually partition the vectors; we use ideas present in our proof to find subfamilies of vectors which may be used to construct a kind of “optimal” partition. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the notion of majorization to , the Banach space of all bounded real sequences, and investigate some of its properties. Considering this notion on , the subspace of all convergent real sequences, the structure of all bounded linear operators which preserve the majorization relation on this subspace is obtained. Finally we introduce two different classes of linear preservers of majorization on which illustrate some important differences between the structure of these operators on and those on spaces, for , as well as those on . [Copyright &y& Elsevier]

Abstract: We consider Jordan derivations of a unital algebra having a nontrivial idempotent. It turns out that on unital algebras there exist Jordan derivations that are not derivations. For this purpose we introduce the term a singular Jordan derivation, which is a proper Jordan derivation of the form that depends on Peirce decomposition of the unital algebra . Singular Jordan derivations are usually antiderivations. The main result of the paper states that under certain conditions every Jordan derivation of is the sum of a derivation and a singular Jordan derivation. [Copyright &y& Elsevier]

Abstract: In this paper we describe the isomorphism classes of finite-dimensional complex Leibniz algebras whose quotient algebra with respect to the ideal generated by squares is isomorphic to the direct sum of three-dimensional simple Lie algebra sl 2 and a three-dimensional solvable ideal. We choose a basis of the isomorphism classes’ representatives and give explicit multiplication tables. [Copyright &y& Elsevier]

Abstract: This paper concerns with the tripotency of a linear combination of three matrices, which has a background in statistical theory. We demonstrate all the possible cases that lead to the tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute. By utilizing block technique and returning the partitioned matrices to their original forms, we derive the sufficient and necessary conditions such that a linear combination of three mutually commuting involutory matrices is tripotent. [Copyright &y& Elsevier]

Abstract: In this paper we characterize the Birkhoff–James orthogonality for elements of a Hilbert C ∗-module in terms of states of the underlying C ∗-algebra. We also show that the Birkhoff–James orthogonality in a Hilbert C ∗-module over a C ∗-algebra and orthogonality with respect to the -valued inner product coincide if and only if is isomorphic to In addition, some new results concerning the case of equality in the triangle inequality for elements of a Hilbert C ∗-module are obtained. [Copyright &y& Elsevier]

Abstract: A -graph is a graph consisting of two cycles and with just a vertex in common. Fan and Luo in Theorem 4.4 alleged that a -graph is determined by its adjacency spectrum if it contains no cycle . However, according to Theorem 6.3 we find that the result in Theorem 4.4 is not completely correct. In this paper, we prove that is DAS if and only if , and has a unique cospectral mate if . [Copyright &y& Elsevier]

Abstract: Let be the set of all matrices with entries in . For , it is said that A is row majorized (respectively column-majorized) by B if every row (respectively column) of A is majorized by the corresponding row (respectively column) of B, i.e. for every i () there exists a doubly stochastic matrix such that , where and are the rows of A and B respectively. In this paper the relations row and column-majorization on are studied and also all linear operators preserving (or strongly preserving) row or column-majorization will be characterized. [Copyright &y& Elsevier]

Abstract: The development of new classes of linearizations of square matrix polynomials that generalize the classical first and second Frobenius companion forms has attracted much attention in the last decade. Research in this area has two main goals: finding linearizations that retain whatever structure the original polynomial might possess, and improving properties that are essential for accurate numerical computation, such as eigenvalue condition numbers and backward errors. However, all recent progress on linearizations has been restricted to square matrix polynomials. Since rectangular polynomials arise in many applications, it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials. In this paper, the family of Fiedler linearizations is extended from square to rectangular matrix polynomials, and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials. Fiedler linearizations are one of the most important classes of linearizations introduced in recent years, but their generalization to rectangular polynomials is nontrivial, and requires a completely different approach to the one used in the square case. To the best of our knowledge, this is the first class of new linearizations that has been generalized to rectangular polynomials. [Copyright &y& Elsevier]

Abstract: Let m and n be positive integers with and . In this paper, the full automorphism group of the generalized unitary graph is determined. [Copyright &y& Elsevier]

Abstract: Let be a simple graph of order , and let be the Laplacian spectrum of . The Laplacian-energy-like invariant of ( for short) is defined as . In this paper, a new lower bound for of graphs in terms of the maximum degree is given. Meanwhile, an upper bound and a lower bound for of the line graph (resp., the subdivision graph and the total graph) of a regular graph are obtained. [Copyright &y& Elsevier]

Abstract: In this paper we demonstrate that there are distinct differences between real and complex equiangular tight frames (ETFs) with regards to erasures. For example, we prove that there exist arbitrarily large non-trivial complex equiangular tight frames which are optimal against three erasures, and that such frames come from a unique class of complex ETFs. In addition, we extend certain results in Bodmann and Paulsen (2005) to complex vector spaces as well as show that other results regarding real ETFs are not valid for complex ETFs. [Copyright &y& Elsevier]

Abstract: It is known that the Schur complements of doubly diagonally dominant matrices are doubly diagonally dominant. In this paper, we obtain an estimate for the doubly diagonally dominant degree on the Schur complement of strictly doubly diagonally dominant matrices. Then, as an application we obtain that the eigenvalues of the Schur complements are located in the Brauer Ovals of Cassini of the original matrices under certain conditions. As another application, we obtain an upper bound for the infinity norm on the inverse on the Schur complement of strictly doubly diagonally dominant matrices. Further, based on the derived results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster. [Copyright &y& Elsevier]

Abstract: This paper develops concepts of controllability and observability for the class of coordinated linear systems, a special class of linear hierarchical systems with a top-to-bottom information structure. The usual concepts of reachability and indistinguishability for linear systems are refined, taking into account the decentralized nature of coordinated linear systems. The corresponding controllability and observability decompositions are derived, and several concepts of controllability and observability for coordinated linear systems are introduced and discussed. These results are then combined to a definition and characterization of stabilizability via dynamic measurement feedback for this class of systems. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the perturbation of a multiple eigenvalue of the Hermitian matrix when it undergoes an off-diagonal perturbation E whose columns have widely varying magnitudes. When some of E’s columns are much smaller than the others, some copies of are much less sensitive than any existing bound suggests. We explain this phenomenon by establishing individual perturbation bounds for different copies of . They show that when is definite the ith bound scales quadratically with the norm of the ith column, and in the indefinite case the bound is necessarily proportional to the product of E’s ith column norm and E’s norm. An extension to the generalized Hermitian eigenvalue problem is also presented. [Copyright &y& Elsevier]

Abstract: The notion of communication complexity seeks to capture the amount of communication between different parties that is required to find the output of a Boolean function when each party is provided with only part of the input. Different variants of the model governing the rules of this communication lead to different connections with problems in combinatorial linear algebra. In particular, problems arise in this context that concern the rank of a -matrix and the minimum rank of a matrix meeting a given combinatorial description. This paper surveys these connections. [Copyright &y& Elsevier]

Abstract: The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for ) is nonzero whenever is an edge in G and is zero otherwise; maximum nullity is taken over the same set of matrices. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. The spread of a graph parameter at a vertex v or edge e of G is the difference between the value of the parameter on G and on or . Rank spread (at a vertex) was introduced in . This paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank/maximum nullity and zero forcing number. Properties of the spreads are established and used to determine values of the minimum rank/maximum nullity and zero forcing number for various types of grids with a vertex or edge deleted. [Copyright &y& Elsevier]

Abstract: Let be the binary Boolean algebra. The Boolean rank, or factorization rank, of a matrix A in is the smallest k such that A can be factored as an times a matrix. The isolation number of a matrix, A, is the largest number of entries equal to 1 in the matrix such that no two ones are in the same row, no two ones are in the same column, and no two ones are in a submatrix of A of the form . It is known that the isolation number of A is always at most the Boolean rank. This paper investigates for each k, if the isolation number of A is k what are some of the possible values of the Boolean rank of A. [Copyright &y& Elsevier]

Abstract: In this paper, we give the eigenvalue decomposition for odd order tridiagonal 2-Toeplitz matrix and derive the explicit expression for integer powers of such matrix. [Copyright &y& Elsevier]

Abstract: A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation. We extend some fundamental concepts from spectral graph theory to complex unit gain graphs. We define the adjacency, incidence and Laplacian matrices, and study each of them. The main results of the paper are eigenvalue bounds for the adjacency and Laplacian matrices. [Copyright &y& Elsevier]

Abstract: The results in this paper are within the scope of matrix pencil completion problems. Given an arbitrary matrix pencil, we obtain necessary and sufficient conditions for the existence of a strictly equivalent pencil with a prescribed constant subpencil for algebraically closed fields. [Copyright &y& Elsevier]

Abstract: In this paper we use doubly stochastic operators to extend the notion of majorization on , where I is assumed to be an infinite set and , and then characterize the structure of all bounded linear maps on this space which preserve majorization. We will see that the two cases and lead to different structures. [Copyright &y& Elsevier]

Abstract: In this paper, some Drazin inverse representations of the linear combinations of two idempotents in a Banach algebra are obtained. Moreover, we present counter-examples to and establish the corrected versions of two theorems by Cvetković-Ilić and Deng. [Copyright &y& Elsevier]

Abstract: In this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its block is not zero and has perturbations. Computable formula of backward errors are derived. The expressions are useful for testing the stability of practical algorithms. [Copyright &y& Elsevier]

Abstract: The main result of this paper is some new equivalent condition for a contraction to be a direct sum of a unitary operator and a completely nonunitary contraction (called Wold-type decomposition). It is applied for -quasihyponormal, -paranormal, -paranormal and of class operators. This decomposition is connected with the Putnam-Fuglede property. Additionally, we show an example of operator of class with norm , but without SVEP (so not paranormal). Finally, we remark that each -paranormal operator is -paranormal. [Copyright &y& Elsevier]

Abstract: Let be a triangular algebra. We say that is a Jordan higher derivable mapping at G if for any with . An element is called a Jordan higher all-derivable point of if every Jordan higher derivable linear mapping at G is a higher derivation. In this paper, under some mild conditions on , we prove that some elements of are Jordan higher all-derivable points. This extends some results in to the case of Jordan higher derivations. [Copyright &y& Elsevier]

Abstract: This paper presents a new approach to computing an approximate solution of Tikhonov-regularized large-scale ill-posed least-squares problems with a general regularization matrix. The iterative method applies a sequence of projections onto generalized Krylov subspaces. A suitable value of the regularization parameter is determined by the discrepancy principle. [Copyright &y& Elsevier]