Sufficient conditions for permutation equivalence to a WHS-matrix

Authors: Eisner, Ludwig¹; Olesky, D. D.²; Driessche, P. van den³

Source: Linear and Multilinear Algebra, Volume 57, Number 1, January 2009 , pp. 103-110(8)

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Abstract:

Square matrices with positive leading principal minors, called WHS-matrices (weak Hawkins-Simon), are considered in economics. Some sufficient conditions for a matrix to be a WHS-matrix after suitable row and/or column permutations have recently appeared in the literature. New and unified proofs and generalizations of some results to rectangular matrices are given. In particular, it is shown that if left multiplication of a rectangular matrix A by some nonnegative matrix is upper triangular with positive diagonal, then some row pemutation of A is a WHS-matrix. For a nonsingular A with either the first nonzero entry of each of its rows positive or the last nonzero entry of each column of A-1 positive, again some row permutation of A is a WHS-matrix. In addition, any rectangular full rank semipositive matrix is shown to be permutation equivalent to a WHS-matrix.

Keywords: LU factorization; Positive leading principal minors; Semipositive matrix; Upper hull; Weak Hawkins-Simon matrix

Document Type: Research article

DOI: http://dx.doi.org/10.1080/03081080701742726

Affiliations: 1: Fakultat fur Mathematik, Universitat Bielefeld, D-33501 Bielefeld, Germany 2: Department of Computer Science, University of Victoria, Victoria, B.C., V8W 3P6, Canada 3: Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., V8W 3P4, Canada

Publication date: 2009-01-01