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For a nonnegative matrix A and real diagonal matrix D, two known inequalitieson the spectral radius, r(A^2 D^2) >= r(AD)^2 and r(A) r(A D^2) >= r(AD)^2,leave open the question of what determines the order of r(A^2 D^2) with respectto r(A) r(A D^2). This is a special case of a broad class of problems thatarise repeatedly in ecological and evolutionary dynamics. Here, sufficientconditions are found on A that determine orders in either direction. For adiagonally symmetrizable nonnegative matrix A with all positive eigenvalues andnonnegative D, r(A^2 D) <= r(A) r(AD). The reverse holds if all of theeigenvalues of A are negative besides the Perron root. This is a particularcase of the more general result that r(A [(1-m) r(B) I + m B] D) is monotonicin m when all non-Perron eigenvalues of A have the same sign --- decreasing forpositive signs and increasing for negative signs, for symmetrizable nonnegativeA and B that commute. Commuting matrices include the Kronecker products A, B in{\otimes_{i=1}^L M_i^{t_i}}, t_i in {0, 1, 2, ...}, which comprise a class ofapplication for these results. This machinery allows analysis of the sign ofd/d m_j r({\otimes_{i=1}^L[(1-m_i) r(A_i) I_i + m_i A_i]}D). The eigenvaluesign conditions also provide lower or upper bounds to the harmonic mean of theexpected sojourn times of Markov chains. These inequalities appear in theasymptotic growth rates of viral quasispecies, models for the evolution ofdispersal in random environments, and the evolution of site-specific mutationrates over the entire genome.

On the Ordering of Spectral Radius Product r(A) r(AD) Versus r(A^2 D) and Related Applications