Sign-Nonsingularity and Orthogonality of Matrices
Dr. Peter Gibson
Department of Mathematical Sciences
University of Alabama in Huntsville
November 11, 1999
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 ATA is a diagonal matrix, and that A is a weighing matrix if ATA = wI for some integer w. Sign-nonsingular column orthogonal matrices and sign-nonsingular weighing matrices are also investigated.
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