On Hadamard powers of positive semi-definite matrices

TL;DR

This paper investigates the structure of the set of exponents for which the Hadamard power of positive semi-definite matrices remains positive semi-definite, revealing complex behaviors including multiple disjoint intervals.

Contribution

It provides new examples of matrices with multiple disjoint intervals in the set of valid Hadamard powers, extending previous understanding of the set structure.

Findings

The set can contain multiple disjoint intervals for certain matrices.

The number of disjoint intervals can grow arbitrarily large with matrix size.

The analysis includes matrices with entries not necessarily non-negative.

Abstract

Consider the set of scalars $\alpha$ for which the $\alpha$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots, n-3\}\cup [n-2, \infty)$. A natural question is "what is the possible form of the set of such $\alpha$ for a fixed p.s.d. matrix with non-negative entries?". In all examples appearing in the literature, the set turns out to be union of a finite set and a semi-infinite interval. In this article, examples of matrices are given for which the set consists of a finite set and more than one disjoint interval of positive length. In fact, it is proved that for some matrices, the number of such disjoint intervals can be made arbitrarily large by taking $n$ large. The case when the entries of the matrices are not necessarily non-negative is also considered.