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(eA) is a log-convex functional over the real diagonal matrices D. He showed that for fully indecomposable A, log r(eA) is strictly convex over D1,D2 if and only if D1 −D2 6= 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>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 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,A2,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.