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On single and double Soules matrices
Authors:Mei Q Chen  Lixing Han
Institution:a Department of Mathematics and Computer Science, The Citadel, Charleston, SC 29409, United States
b Department of Mathematics, University of Michigan—Flint, Flint, MI 48502, United States
c Department of Mathematics, University of Connecticut, 196 Auditorium Road, Unit 3009, Storrs, CT 06269-3009, United States
Abstract:Until now the concept of a Soules basis matrix of sign patternN consisted of an orthogonal matrix RRn,n, generated in a certain way from a positive n-vector, which has the property that for any diagonal matrix Λ = diag(λ1, … , λn), with λ1 ? ? ? λn ? 0, the symmetric matrix A = RΛRT has nonnegative entries only. In the present paper we introduce the notion of a pair of double Soules basis matrices of sign patternN which is a pair of matrices (PQ), each in Rn,n, which are not necessarily orthogonal and which are generated in a certain way from two positive vectors, but such that PQT = I and such that for any of the aforementioned diagonal matrices Λ, the matrix A = PΛQT (also) has nonnegative entries only. We investigate the interesting properties which such matrices A have.As a preamble to the above investigation we show that the iterates, View the MathML source, generated in the course of the QR-algorithm when it is applied to A = RΛRT, where R is a Soules basis matrix of sign pattern N, are again symmetric matrices generated by the Soules basis matrices Rk of sign pattern N which are themselves modified as the algorithm progresses.Our work here extends earlier works by Soules and Elsner et al.
Keywords:Nonnegative matrices  M-matrices  The inverse eigenvalue problem  QR-algorithm  Soules bases
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