Non-Singularity Conditions for Two Z-Matrix Types ()
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.
Share and Cite:
Miura, S. (2014) Non-Singularity Conditions for Two Z-Matrix Types.
Advances in Linear Algebra & Matrix Theory,
4, 109-119. doi:
10.4236/alamt.2014.42009.