The inverse C = [ci,j ] of an irreducible nonsingular symmetric tridiagonal matrix is a socalled Green’s matrix, i.e. it is given by two sequences of real numbers {ui} and {vi} such that ci,j = uivj for i ≤ j. A similar result holds for nonsymmetric matrices. An open problem on nonsingular sparse matrices is whether there exists a similar structure for their inverses as in the tridiagonal case....

It is well known that any nonsingular M–matrix admits an LU factorization into M–matrices (with L and U lower and upper triangular respectively) and any singular M–matrix is permutation similar to an M–matrix which admits an LU factorization into M–matrices. Varga and Cai establish necessary and sufficient conditions for a singular M–matrix (without permutation) to allow an LU factorization wit...

In a recent paper Holtz showed that M–matrices satisfy Newton’s inequalities and so do the inverses of nonsingular M–matrices. Since nonsingular M–matrices and their inverses display various types of monotonic behavior, monotonicity properties adapted for Newton’s inequalities are examined for nonsingular M–matrices and their inverses. In the second part of the paper the problem of whether Draz...

A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang (2001) showed that the inertia of a signnonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such matrices. The algorithm runs in O(nm) time ...

Abstract This paper characterizes a class of regular para-Hermitian transfer matrices and then studies the J-spectral factorization of this class using similarity transformations. A transfer matrix Λ in this class admits a J-spectral factorization if and only if there exists a common nonsingular matrix to similarly transform the A-matrices of Λ and Λ, resp., into 2 × 2 lower (upper, resp.) tria...

Many algorithms for Independent Component Analysis rely on a simultaneous diagonalization of a set of matrices by means of a nonsingular matrix. In this paper we provide means to determine the matrix when it has more columns than rows.

Two Hermitian matrices A, B ∈ Mn(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C ∈ Mn(C) such that B = CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible iner...

An explicit formula for the polar decomposition of an n n nonsingular companion matrix is derived. The proof involves the largest and smallest singular values of the companion matrix.