M-matrix and inverse M-matrix extensions

Linear Algebra and its Applications, 2008

Generalizations of M-matrices are studied, including the new class of GM-matrices. The matrices studied are of the form sI − B with B having the Perron-Frobenius property, but not necessarily being nonnegative. Results for these classes of matrices are shown, which are analogous to those known for M-matrices. Also, various splittings of a GM-matrix are studied along with conditions for their convergence.

The Electronic Journal of Linear Algebra

In this article, some new results on $M$-matrices, $H$-matrices and their inverse classes are proved. Specifically, we study when a singular $Z$-matrix is an $M$-matrix, convex combinations of $H$-matrices, almost monotone $H$-matrices and Cholesky factorizations of $H$-matrices.

Linear Algebra and its Applications, 1987

The concept of a singular M-matrix A with respect to a proper cone X is extended, by replacing the usual regularity condition A = al -B for a Xnonnegative matrix B with the weaker condition, exponential nonnegativity of -A. As in earlier work which dealt with the nonsingular case, in the present characterizations the lack of regularity is overcome by employing subtangentiality.

Electronic Journal of Linear Algebra, 2012

A well-known property of an irreducible singular M -matrix is that it has a generalized inverse which is non-negative, but this is not always true for any generalized inverse. The authors have characterized when the Moore-Penrose inverse of a symmetric, singular, irreducible and tridiagonal M -matrix is itself an M -matrix. We aim here at giving new explicit examples of infinite families of matrices with order up to 4 having this property.

2008

In this paper, we will study algebraic properties of the gen- eralized inverses of the sum and the product of two matrices. By using rank additivity we explicit the generalized inverse of the sum of two matrices if their range spaces are not disjoint and we give a numerical example in this case. We will also use projectors to express the general form of a generalized inverse of the product of two matrices.

Linear Algebra and its Applications, 2012

A well-known property of an irreducible singular M-matrix is that it has a generalized inverse which is non-negative, but this is not always true for any generalized inverse. The authors have characterized when the Moore-Penrose inverse of a symmetric, singular, irreducible and tridiagonal M-matrix is itself an M-matrix. We aim here at giving new explicit examples of infinite families of matrices with order up to 4 having this property.

Linear Algebra and its Applications, 1977

The purpose of this survey is to classify systematically a widely ranging list of characterizations of nonsingular M-matrices from the economics and mathematics literatures. These characterizations are grouped together in terms of their relations to the properties of (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability and (4) semipositivity and diagonal dominance. A list of forty equivalent conditions is given for a square matrix A with nonpositive off-diagonal entries to be a nonsingular M-matrix. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. In addition, other remarks relating nonsingular L-matrices to certain complex matrices are made, and the recent literature on these general topics is surveyed.

World Scientific News, 2018

We analyze the compatibility of linear systems and the uniqueness of the corresponding solution via the singular value decomposition of an arbitrary n x m matrix. Our approach leads in natural manner to the Moore-Penrose's generalized inverse between the subspaces of activation of the matrix under study.

Applied Mathematics and Computation, 2014

The purpose of this paper is to introduce a new generalized inverse, called DMP inverse, associated with a square complex matrix using its Drazin and Moore-Penrose inverses. DMP inverse extends the notion of core inverse, introduced by O.M. Baksalary and G. Trenkler for matrices of index at most 1 in [Core inverse of matrices, Linear and Multilinear Algebra, 2010, 681-697] to matrices of an arbitrary index. DMP inverses are analyzed from both algebraic as well as geometrical approaches establishing the equivalence between them.

Linear Algebra and its Applications, 2009

Generalized inverse of matrix Matrix equality Additive decomposition Block decomposition Parallel sum of matrices Shorted matrix Schur complement Matrix rank method Necessary and sufficient conditions are derived for the matrix equality A − = PN − Q to hold, where A − and N − are generalized inverses of matrices. Some consequences and applications are also given. In particular, necessary and sufficient conditions are derived for the additive decompositions C