Positivity properties of some special matrices and their Hadamard powers

Abstract

The r-th Hadamard (or entrywise) power of a nonnegative positive semidefinite matrix need not be positive semidefinite for all positive real numbers r. The problems of studying the positive definiteness (or positive semidefiniteness) and total positivity (or total nonnegativity) of Hadamard powers of a matrix or a family of matrices have been of tremendous interest in matrix theory. An entrywise nonnegative matrix is called infinitely divisible if its r-th Hadamard power is positive semidefinite for every rgt0. For positive real numbers $\la_1lt\cdotslt\la_n$, we consider the matrix $\left[\frac{1}{\beta(\la_i,\la_j)}\right]$, where $\beta(.,.)$ denotes the beta function. We shall present our work on its infinite divisibility and the total positivity of its Hadamard powers. We shall give an important decomposition known as a successive elementary bidiagonal decomposition for the beta matrix $\mathcal{B}=\left[\frac{1}{\beta(i, j)}\right]$. We consider a few band matrices and identify all the Hadamard powers preserving the positive (semi) definiteness. We shall discuss similar results for a few more special matrices, namely, Bell matrices, Stirling matrices, characteristic matrices, and mean matrices. The matrix $S = [1+x_i y_j]$, where $0ltx_1lt\cdotsltx_n,\, 0lty_1lt\cdotslty_n$, has gained importance lately due to its role in powers preserving total nonnegativity. We shall give an explicit decomposition of $S$ in terms of elementary bidiagonal matrices. newline