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

TL;DR

This paper investigates inequalities involving the spectral radius of nonnegative matrices and diagonal matrices, providing conditions that determine the order of spectral radii and applying these results to ecological, evolutionary, and Markov chain models.

Contribution

It establishes new sufficient conditions for the ordering of spectral radii in products involving nonnegative matrices and diagonal matrices, extending to applications in biological and Markov chain models.

Findings

For diagonally symmetrizable matrices with positive eigenvalues, r(A^2 D) <= r(A) r(AD).

Eigenvalue sign conditions determine the monotonicity of spectral radius in matrix combinations.

Results apply to models of viral growth, dispersal evolution, and mutation rates.

Abstract

For a nonnegative matrix A and real diagonal matrix D, two known inequalities on 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 respect to r(A) r(A D^2). This is a special case of a broad class of problems that arise repeatedly in ecological and evolutionary dynamics. Here, sufficient conditions are found on A that determine orders in either direction. For a diagonally symmetrizable nonnegative matrix A with all positive eigenvalues and nonnegative D, r(A^2 D) <= r(A) r(AD). The reverse holds if all of the eigenvalues of A are negative besides the Perron root. This is a particular case of the more general result that r(A [(1-m) r(B) I + m B] D) is monotonic in m when all non-Perron eigenvalues of A have the same sign --- decreasing for positive signs and increasing for negative signs, for symmetrizable nonnegative A 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 of application for these results. This machinery allows analysis of the sign of d/d m_j r({\otimes_{i=1}^L[(1-m_i) r(A_i) I_i + m_i A_i]}D). The eigenvalue sign conditions also provide lower or upper bounds to the harmonic mean of the expected sojourn times of Markov chains. These inequalities appear in the asymptotic growth rates of viral quasispecies, models for the evolution of dispersal in random environments, and the evolution of site-specific mutation rates over the entire genome.