Orthogonal bases that leads to symmetric nonnegative matrices
Interpreting inequalities for matrices elementwise, then a matrix \(R\in{\mathbb{R}}^{n\times n}\) is called a Soules matrix if (1) it is orthogonal, (2) its first column \(r_1\geq0\), (3) for all diagonal matrices \(\Lambda\geq 0\) with nonincreasing diagonal elements, \(S=R\Lambda R^T\geq0\). A matrix \(A\) is called an M-matrix if it is of the form \(cI-G\) with \(G\geq0\) and \(c\geq\) the spectral radius of \(G\). If \(A^k\) is an irreducible M-matrix for all \(k\geq1\), then \(A\) is called an MMA-matrix. A matrix \(A\) is called an inverse MMA-matrix if its inverse is an MMA-matrix. This paper shows how to construct all Soules matrices, starting from its first column \(r_1\). Furthermore, it is proved that the class of matrices \(S\) of the above form coincides with the class of symmetric inverse MMA-matrices and that the class of nonsingular such matrices \(S\) can be mapped onto another class which is up to a conjugation by positive diagonal matrices the so called class of strictly ultrametric matrices. Finally, a special sign pattern for the elements of \(Q\) and \(R\) in the QR factorization of certain such matrices \(S\) is explained.
scientific article; zbMATH DE number 1116058