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

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.