Double Soules pairs and matching Soules bases
This paper deals with double Soules pairs and bases. A Soules basis matrix is an orthogonal matrix \(R\) with a positive first column, such that \(R\Lambda R^T\) is nonnegative for every nonnegative diagonal matrix \(\Lambda\) with nonincreasing diagonal elements [\textit{G. W. Soules}, Linear Multilinear Algebra 13, 241--251 (1983; Zbl 0516.15013)]. In the same paper, for a positive unit vector, a Soules basis matrix with that vector as its first column is constructed. Later, in \textit{L. Elsner, R. Nabben} and \textit{M. Neumann} [Linear Algebra Appl. 271, 323--343 (1998; Zbl 0891.15019)] described all \(n\times n\) matrices of this type and considered Soules matrices. Afterwards, \textit{M. Q. Chen, L. Han} and \textit{M. Neumann} [Linear Algebra Appl. 416, No.~1, 88--110 (2006; Zbl 1109.15015)] introduced the notion of a double Soules pair, providing an algorithm for generating such pairs of matrices. In the current paper the authors show how to construct all double Soules pairs and properties of the nonnegative matrices generated by them are studied. In addition, the authors consider a second generalization, namely a pair of matching Soules bases. Characterizations are given, and properties of the generated generalized inverses are discussed.
scientific article; zbMATH DE number 5132599