Non-Singularity Conditions for Two Z-Matrix Types

Shinji Miura

Independent, Gifu, Japan.

DOI: 10.4236/alamt.2014.42009                

Abstract

A real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. This paper shows a necessary and sufficient condition for non-singularity of two types of Z-matrices. The first is for the Z-matrix whose row sums are all non-negative. The non-singularity condition for this matrix is that at least one positive row sum exists in any principal submatrix of the matrix. The second is for the Z-matrix which satisfies where . Let be the ith row and the jth column element of , and be the jth element of . Let be a subset of which is not empty, and be the complement of if is a proper subset. The non-singularity condition for this matrix is such that or such that for  . Robert Beauwens and Michael Neumann previously presented conditions similar to these conditions. In this paper, we present a different proof and show that these conditions can be also derived from theirs.

Keywords

Z-Matrix, M-Matrix, Non-Negative Matrix, Diagonal Dominance

Conflicts of Interest

The authors declare no conflicts of interest.

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