Positivity of Hadamard powers of a few band matrices

TL;DR

This paper characterizes the powers that preserve positive semidefiniteness and positive definiteness of band matrices constrained by graphs, providing explicit descriptions and analyzing infinite divisibility for specific matrix families.

Contribution

It offers a combinatorial characterization of the critical exponent for positive definiteness on graph-constrained matrices and identifies the powers preserving positive (semi) definiteness for certain band matrices.

Findings

Critical exponent for positive definiteness is explicitly described for all chordal graphs.

Powers r ≥ 1 preserve positive definiteness of all tridiagonal matrices with nonnegative entries.

Results extend to specific pentadiagonal matrices and analyze their infinite divisibility.

Abstract

Let $\mathbb{P}_G([0,\infty))$ and $\mathbb{P}_G^{'}([0,\infty))$ be the sets of positive semidefinite and positive definite matrices of order $n$, respectively, with nonnegative entries, where some positions of zero entries are restricted by a simple graph $G$ with $n$ vertices. It is proved that for a connected simple graph $G$ of order $n\geq 3$, the set of powers preserving positive semidefiniteness on $\mathbb{P}_G([0,\infty))$ is precisely the same as the set of powers preserving positive definiteness on $\mathbb{P}_G^{'}([0,\infty))$. In particular, this provides an explicit combinatorial description of the critical exponent for positive definiteness, for all chordal graphs. Using chain sequences, it is proved that the Hadamard powers preserving the positive (semi) definiteness of every tridiagonal matrix with nonnegative entries are precisely $r\geq 1$. The infinite divisibility of tridiagonal matrices is studied. The same results are proved for a special family of pentadiagonal matrices.