Double Soules pairs and matching Soules bases

We consider two generalizations of the notion of a Soules basis matrix. A pair of nonsingular n × n matrices (P,Q) is called a double Soules pair if the first columns of P and Q are positive, PQT = I, and PΛQT is nonnegative for every n × n nonnegative diagonal matrix Λ with nonincreasing diagonal elements. In a paper by Chen, Han, and Neumann an algorithm for generating such pairs was given. Here we characterize all such pairs, and discuss some implications of the characterization. We also consider pairs of matrices (U,V) such that U and V each consists of k orthonormal columns, the first of which is positive, and UΛV is nonnegative for every k × k nonnegative diagonal matrix Λ with nonincreasing diagonal elements. We call such pairs matching Soules pairs. We characterize all such pairs, and make some observations regarding the nonnegative matrices and generalized inverses generated by them.