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.


Friedland (1981) showed that for a nonnegative square matrix A, the spectral radius r(eD A) is a log-convex functional over the real diagonal matrices D, and that when A is fully indecomposable, log r(eD A) is strictly convex over a convex combination of D1, D2 if and only if D1 - D2 ≠ c I for any c ∈ R. Here the condition of full indecomposability is shown to be replaceable by the weaker condition that A and ATA be irreducible, which is the sharpest possible replacement condition. Irreducibility of both A and ATA is shown to be equivalent to irreducibility of A2 and ATA, 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, ATA, and A AT, 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.
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