The critical exponent conjecture for powers of doubly nonnegative matrices

TL;DR

This paper proves the exact critical exponent for powers of doubly nonnegative matrices, confirming a recent conjecture and extending the results to broader classes of functions with multiple approaches.

Contribution

It establishes the precise critical exponent for continuous powers of doubly nonnegative matrices and generalizes the conjecture to broader functional classes.

Findings

Critical exponent for doubly nonnegative matrices is exactly n-2.

All continuous powers beyond n-2 preserve doubly nonnegative property.

The conjecture is confirmed and extended to broader function classes.

Abstract

Doubly non-negative matrices arise naturally in many setting including Markov random fields (positively banded graphical models) and in the convergence analysis of Markov chains. In this short note, we settle a recent conjecture by C.R. Johnson et al. [Linear Algebra Appl. 435 (2011)] by proving that the critical exponent beyond which all continuous conventional powers of $n$-by-$n$ doubly nonnegative matrices are doubly nonnegative is exactly $n-2$. We show that the conjecture follows immediately by applying a general characterization from the literature. We prove a stronger form of the conjecture by classifying all powers preserving doubly nonnegative matrices, and proceed to generalize the conjecture for broad classes of functions. We also provide different approaches for settling the original conjecture.