A sharpened condition for strict log-convexity of the spectral radius via the bipartite graph

Friedland (1981) [10], showed that for a nonnegative square matrix A, the spectral radius is a log-convex functional over the real diagonal matrices D. He showed that for fully indecomposable is strictly convex over D1,D2 if and only if for any c∈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 A2 and A⊤A, which is the condition for a number of strict inequalities on the spectral radius found in Cohen et al. (1982) [8]. Such ‘two-fold irreducibility’ is equivalent to joint irreducibility of , 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.