Solutions of Z-matrix equations

Solutions of Z-Matrlx Equations'"

We investigate the existence and the nature of the solutions of the matrix equation Ax = b, where A is a Z-matrix and b is a nonnegative vector. When x is required to be nonnegative, then an existence theorem is due to Carlson and Victory and is re-proved in this paper. We apply our results to study nonnegative vectors in the range of Z-matrices.

Least Squares Solutions of Inconsistent Fuzzy Linear Matrix Equations

Fast verification of solutions of matrix equations

In this paper, we are concerned with a matrix equation Ax = b where A is an n× n real matrix and x and b are n-vectors. Assume that an approximate solution e x is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for ‖A−1b − e x‖∞. The emphasis is on rigour of the bounds. The purpose of ...

Positive definite solutions of some matrix equations

In this paper we investigate some existence questions of positive semi-definite solutions for certain classes of matrix equations known as the generalized Lyapunov equations. We present sufficient and necessary conditions for certain equations and only sufficient for others. © 2008 Elsevier Inc. All rights reserved. AMS classification: 15A24; 15A48; 42A82; 47A62

Accurate solutions of M-matrix Sylvester equations

This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an M -matrix Sylvester equation AX + XB = C by which we mean both A and B have positive diagonal entries and nonpositive off-diagonal entries and P = Im⊗A+B⊗ In is a nonsingular M -matrix, and C is entrywise nonnegative. It is proved that small relative perturbations to the e...