On single and double Soules matrices

Until now the concept of a Soules basis matrix of sign pattern N consisted of an orthogonal matrix R∈Rn,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 pattern N which is a pair of matrices (P, Q), 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, , 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.