Generalized $M$-matrices and applications

Linear and Multilinear Algebra, 2016

The well-known classes of EP matrices and normal matrices are defined by the matrices that commute with their Moore-Penrose inverse and with their conjugate transpose, respectively. This paper investigates the class of m-EP matrices and m-normal matrices that provide a generalization of EP matrices and normal matrices, respectively, and analyzes both of them for their properties and characterizations.

The word "matrix" comes from the Latin word for "womb" because of the way that the matrix acts as a womb for the data that it holds. The first known example of the use matrices was found in a Chinese text called Nine Chapters of the Mathematical Art, which is thought to have originated somewhere between 300 B.C. and 200 A.D. The modern method of matrix solution was developed by a German mathematician and scientist Carl Friedrich Gauss. There are many different types of matrices used in different modern career fields. We introduce and discuss the different types of matrices that play important roles in various fields.

Linear Algebra and its Applications, 2006

In this paper we study conditions under which a specially structured Z-matrix is an M-matrix. We apply the result to a capacity problem in wireless communications.

AL-Rafidain Journal of Computer Sciences and Mathematics, 2011

In this work we present a subset of M  is the set of all n n matrices ) which we called the set of special matrices and denoted it by n n S  . We give some important properties of n n S  .

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.

Linear Algebra and its Applications, 1988

A proof is given for the preferred basis theorem for the generalized nullspace of a given M-matrix. The theorem is then generalized and extended to the case of a Z-matrix. In this paper we give a proof for a known result, namely, the preferred basis theorem for the generalized nullspace of a given M-matrix. We then generalize and extend the theorem to the case of a Z-matrix. Let A be a Z-matrix. A preferred set is an ordered set of semipositive vectors such that the image of each vector under -A is a nonnegative linear combination of the subsequent vectors. Furthermore, the positivity of the *

Linear Algebra and its Applications, 2012

H−matrices play an important role in the theory and applications of Numerical Linear Algebra. So, it is very useful to know whether a given matrix A ∈ C n,n , usually the coefficient of a complex linear system of algebraic equations or of a Linear Complementarity Problem (A ∈ R n,n , with a ii > 0 for i = 1, 2,. .. , n in this case), is an H−matrix; then, most of the classical iterative methods for the solution of the problem at hand converge. In recent years the set of H−matrices has been extended to what is now known as the set of General H−matrices, and a partition of this set in three different classes has been made. The main objective of this work is to develop an algorithm that will determine the H−matrix character and will identify the class to which a given matrix A ∈ C n,n belongs; in addition, some results on the classes of general H−matrices and a partition of the non-H−matrix set are presented.

Electronic Journal of Linear Algebra, 2009

An M ∨ -matrix has the form A = sI -B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., B k is entrywise nonnegative for all sufficiently large integers k. A theory of M ∨ -matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.

Proceedings of the American Mathematical Society, 1983

We consider the sets of "singular" matrices corresponding to generalized matrix functions. The latter are generalizations of the determinant function. It is shown that two generalized matrix functions determine the same set of "singular" matrices if and only if they differ by a scalar multiple.