On single and double Soules matrices

Until now the concept of a Soules basis matrix of sign pattern consisted of an orthogonal matrix , 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 which is a pair of matrices (P, Q), each in , 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 , are again symmetric matrices generated by the Soules basis matrices Rk of sign pattern which are themselves modified as the algorithm progresses. Our work here extends earlier works by Soules and Elsner et al.