Bidiagonal decompositions and total positivity of some special matrices

Abstract

The matrix $S=[1+x_i{y}_j{]}_{i,j=1}^n,0<x_1<\cdots <x_n,0<y_1<\cdots <y_n$, has gained importance lately due to its role in powers preserving total nonnegativity. We give an explicit decomposition of $S$ in terms of elementary bidiagonal matrices, which is analogous to the Neville decomposition. We give a bidiagonal decomposition of $S^{\circ m}=[(1+x_i{y}_j{)}^m]$ for positive integers $1\le m\le n-1$. We also explore the total positivity of Hadamard powers of another important class of matrices called mean matrices.

arXiv commentThe article contains 15 pages. It has been accepted in Operators and Matrices