The critical exponent for continuous conventional powers of doubly nonnegative matrices

TL;DR

This paper investigates the critical exponent for continuous powers of doubly nonnegative matrices, establishing bounds and conjecturing its exact value, with proofs for specific matrix sizes and cases.

Contribution

It introduces the concept of a critical exponent for continuous powers of doubly nonnegative matrices and provides bounds and partial proofs for its exact value.

Findings

Critical exponent is at least n-2.

Conjecture that the critical exponent equals n-2.

Proved the conjecture for n<6 and certain cases.

Abstract

We prove that there exists an exponent beyond which all continuous conventional powers of n-by-n doubly nonnegative matrices are doubly nonnegative. We show that this critical exponent cannot be less than $n-2$ and we conjecture that it is always $n-2$ (as it is with Hadamard powering). We prove this conjecture when $n<6$ and in certain other special cases. We establish a quadratic bound for the critical exponent in general.