On single and double Soules matrices

Abstract

Until now the concept of a Soules basis matrix of sign pattern N consisted of an orthogonal matrix R ∈ R n , n , generated in a certain way from a positive n-vector, which has the property that for any diagonal matrix Λ = diag( λ 1, … , λ n ), with λ 1 ⩾ ⋯ ⩾ λ n ⩾ 0, the symmetric matrix A = RΛR T has nonnegative entries only. In the present paper we introduce the notion of a pair of double Soules basis matrices of sign pattern N which is a pair of matrices ( P, Q), each in R n , n , which are not necessarily orthogonal and which are generated in a certain way from two positive vectors, but such that PQ T = I and such that for any of the aforementioned diagonal matrices Λ, the matrix A = PΛQ T (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, A k = R k Λ R k T , generated in the course of the QR-algorithm when it is applied to A = RΛR T, where R is a Soules basis matrix of sign pattern N , are again symmetric matrices generated by the Soules basis matrices R k of sign pattern N which are themselves modified as the algorithm progresses. Our work here extends earlier works by Soules and Elsner et al.