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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]

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]

A connected graph G , all of whose blocks are cycles or cliques, is called a cycle–clique graph. Let D be the distance matrix of G . By a theorem of Graham et al., we have det ( D ) ≠ 0 if all cycle blocks have odd vertices. In this paper we give the formula for the inverse of D . [ABSTRACT FROM AUTHOR]

Abstract: In [S.K. Hwang, S.S. Pyo, The inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 379 (2004) 77–83] it was claimed that: if and , then there is a symmetric positive doubly stochastic matrix A with the eigenvalues . Afterwards, Fang [M.Z. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 432 (2010) 2925–2927] presented a counterexample to demonstrate that the above proposition was inaccurate. However, the author did not give a solution for a real n-tuple to be the spectrum of a symmetric positive doubly stochastic matrix of order n. In this paper, we give some sufficient conditions to make up for this deficiency. [Copyright &y& Elsevier]

Abstract: In this paper, we shall examine an additive problem of existence and presentation of in terms of D-invertible ring elements a and b and their Drazin inverses and we shall study the existence of the Drazin inverse of a two by two matrix with zero entry. We partly solved a problem posed by Patrício and Hartwig (2012) 22. [Copyright &y& Elsevier]

Abstract: The paper serves as a correction to J. Math. Anal. Appl. 359 (2009) 731–738 dealing with the Drazin invertibility of combinations of idempotents in a Banach algebra. As the arguments employed to do this work equally well for both the usual and generalized Drazin inverse, the latter is included in the discussion which covers the equivalence of the Drazin invertibility of , , and in general of with , as well as the equivalence of the Drazin invertibility of the commutator and anticommutator of . [Copyright &y& Elsevier]

Abstract: In this paper we consider a generalized inverse eigenvalue problem , where is a Jacobi matrix and is a nonsingular diagonal matrix that may be indefinite. Let be leading principal submatrix of . Given , two vectors , , two distinct real numbers , , we construct a Jacobi matrix and two vectors , such that , and , where and . We find necessary and sufficient conditions for solvability of this problem and we give an example. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate stable perturbations and their characterizations for various types of outer inverses, such as generalized, -, -, -, -, -, Moore–Penrose, group, Drazin and generalized Drazin inverses. Some known results are improved and extended. [Copyright &y& Elsevier]

Abstract: The sign pattern of a real matrix M is the -matrix obtained from M by replacing each entry by its sign. Let be the set of real matrices with the same sign pattern as M. For any , if the Drazin inverses of M and have the same sign pattern, then M is said to have signed Drazin inverse. In this paper, we give a complete characterization for a class of anti-triangular matrices with signed Drazin inverse, and present a complete characterization for sign symmetric bipartite matrices with signed Drazin inverse. [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: In this paper, we investigate the properties of a special kind of periodic Jacobi matrices. We show that the solution of the inverse problem for periodic Jacobi matrices is unique if and only if the matrix is of that special kind. Moreover, we present an algorithm to construct the solution of that inverse problem and give some illustrative examples. Finally, we perform a stability analysis of the algorithm. [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: 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, for a given set of numbers with special conditions, we construct a nonnegative matrix such that the given set is its spectrum. [Copyright &y& Elsevier]

Abstract: The inverse mean first passage time problem is given a positive matrix , then when does there exist an n-state discrete-time homogeneous ergodic Markov chain , whose mean first passage matrix is M? The inverse M-matrix problem is given a nonnegative matrix A, then when is A an inverse of an M-matrix. The main thrust of this paper is to show that the existence of a solution to one of the problems can be characterized by the existence of a solution to the other. In so doing we extend earlier results of Tetali and Fiedler. [Copyright &y& Elsevier]

Abstract: In this paper we prove the formula for the expression in terms of , assuming some conditions for and . Here denotes the generalized -weighted Drazin inverse of a linear bounded operator on a Banach space. [Copyright &y& Elsevier]

Abstract: Throughout the last decades, several results have been published in the area of the so-called Matrix Completion Problems. In this paper, we survey several results in this field. In particular, we describe the possible eigenvalues, the characteristic polynomial, the invariant polynomials, or the number of nontrivial invariant polynomials of a square matrix, over a field, when some of its entries are prescribed and the others vary. Finally, we present our contribution, generalizing some of the previous cases, to an matrix partitioned into blocks, with entries in a field, when some of its blocks are prescribed and the others vary. [Copyright &y& Elsevier]

Abstract: In Hwang and Pyo [S.G. Hwang, S.S. Pyo, The inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 379 (2004) 77–83], it is claimed that : for a real -tuple with , ifthen there exists a symmetric positive doubly stochastic matrix whose spectrum is . In this paper we give a counterexample to this proposition. [Copyright &y& Elsevier]