The critical exponent for generalized doubly nonnegative matrices

TL;DR

This paper investigates the critical exponent for generalized doubly nonnegative matrices, extending known results from doubly nonnegative matrices and revealing new bounds and properties for this broader class.

Contribution

It establishes the existence of a critical exponent for GDN matrices, provides bounds, and compares it to the DN case, highlighting differences such as the absence of a CE for Hadamard powers.

Findings

Critical exponent exists for GDN matrices.

Bounded the CE with a quadratic function.

No CE for continuous Hadamard powers of GDN matrices.

Abstract

It is known that the critical exponent (CE) for conventional, continuous powers of $n$-by-$n$ doubly nonnegative (DN) matrices is $n-2$. Here, we consider the larger class of diagonalizable, entry-wise nonnegative $n$-by-$n$ matrices with nonnegative eigenvalues (GDN). We show that, again, a CE exists and are able to bound it with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for $n=3$. There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for continuous Hadamard powers of GDN matrices, despite it also being $n-2$ for DN matrices.