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.
Highlights
A real square matrix whose non-diagonal elements are non-positive is called a Z-matrix
Theorem 2.4 If there exists at least one principal submatrix of an NSZ-matrix whose row sums are all zeroes, 6Cf
If all row sums of A itself, which is one of the principal submatrices of an NSZ-matrix, are zeroes, the proposition is derived from Theorem 1.5 immediately
Summary
A real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. We denote this as a Non-negative Sums Z-matrix (NSZ-matrix). Theorem 1.1 An NSZ-matrix is equivalent to an NPZ-matrix where all elements of x are the same number. ( ) ( ) ∑ ∑ If A = aij is an NSZ-matrix, the ith element of Ax is a j∈N ij x* . ( ) ( ) consider that A = aij is an NPZ-matrix which satisfies ∑ a j∈N ij x* ≥ 0 for ∀i ∈ N where ( ) ∑ ∑ ∃x* > 0. Theorem 1.4 An M-matrix A is non-singular if and only if λ ( B) < ρ In this case, ρ > 0 and all elements of the inverse of an M-matrix are non-negative.
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