A Sharpened Condition for Strict Log-Convexity of the Spectral Radius via the Bipartite Graph

TL;DR

This paper refines conditions for strict log-convexity of the spectral radius of nonnegative matrices, replacing full indecomposability with weaker irreducibility conditions related to the matrix and its transpose, with implications for graph connectivity.

Contribution

It establishes that irreducibility of A and A'A suffices for strict log-convexity, weakening the previously required full indecomposability condition, and links this to graph connectivity properties.

Findings

Irreducibility of A and A'A is sufficient for strict log-convexity.

Two-fold irreducibility is equivalent to certain spectral inequalities.

Graph connectivity characterizes the irreducibility conditions.

Abstract

Friedland (1981) showed that for a nonnegative square matrix A, the spectral radius r(e^D A) is a log-convex functional over the real diagonal matrices D. He showed that for fully indecomposable A, log r(e^D A) is strictly convex over D_1, D_2 if and only if D_1-D_2 != c I for any c \in R. Here the condition of full indecomposability is shown to be replaceable by the weaker condition that A and A'A be irreducible, which is the sharpest possible replacement condition. Irreducibility of both A and A'A is shown to be equivalent to irreducibility of A^2 and A'A, which is the condition for a number of strict inequalities on the spectral radius found in Cohen, Friedland, Kato, and Kelly (1982). Such `two-fold irreducibility' is equivalent to joint irreducibility of A, A^2, A'A, and AA', or in combinatorial terms, equivalent to the directed graph of A being strongly connected and the simple bipartite graph of A being connected. Additional ancillary results are presented.