Sign-Nonsingularity and Orthogonality of Matrices

Dr. Peter Gibson

Department of Mathematical Sciences
University of Alabama in Huntsville

November 11, 1999

Abstract

A real matrix A is said to be sign-nonsingular if each matrix with the same sign pattern as A is nonsingular. Properties of sign-nonsingular orthogonal matrices are developed. One of these shows that for integers k andn, n ≥ 2, there exists an indecomposable, sign-nonsingular orthogonal matrix of order n with exactly k nonzero entries if and only if 4n − 4 ≤ k ≤ (n² + 3n − 2) / 2. Let A be a (0, 1, −1)-matrix of order n. We say thatA is column orthogonal if A^TA is a diagonal matrix, and that A is a weighing matrix if A^TA = wI for some integer w. Sign-nonsingular column orthogonal matrices and sign-nonsingular weighing matrices are also investigated.