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

### Lee Altenberg

2013. *Linear Algebra and Its Applications* 438 (9): 3702-3718. Submitted August 5, 2012.

#### 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*, and that when *A* is fully indecomposable, *log r(e*^{D} A) is strictly convex over a convex combination of *D*_{1}, D_{2} if and only if *D*_{1} - D_{2} ≠ c I for any *c ∈ R*. Here the condition of full indecomposability is shown to be replaceable by the weaker condition that *A* and *A*^{T}A be irreducible, which is the sharpest possible replacement condition. Irreducibility of both *A* and *A*^{T}A is shown to be equivalent to irreducibility of *A*^{2} and *A*^{T}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^{T}A, and *A A*^{T}, 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.

Lee Altenberg's Home Page |
Papers |
E-mail me