ON THE ORDERING OF SPECTRAL RADIUS PRODUCT r(A)r(AD) VERSUS r(A~2D) AND RELATED APPLICATIONS

摘要

For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(A~2D~2) > r(AD)~2 and r(A)r(AD~2)≥ r(AD)~2, leave. open the questio of what determines the order of r(A~2D~2) with respect to r(A)r(AD~2). This is a special case of broad class of problems that arise repeatedly in ecological and evolutionary dynamics. Here, sufficien 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(à~2D) ≤r(A) r(AD). The reverse holds if all of the eigenvalues of A are negative besides the Perron root. This is particular case of the more general result that r(A[(l-m) r(B) I + mB]D) is monotonic in m whe all non-Perron eigenvalues of A have the same sign-decreasing for positive signs and increasing fc negative signs, for symmetrizable nonnegative A and B that commute. Commuting matrices includ the Kronecker products A, B ∈{?L_i=1~M_i~(ii)}, t_i ∈ {0,1,2,...}, which comprise a class for applicatio of these results. This machinery allows analysis of the sign of ?/?m_j r({?_i~L=1[(l-m_i).r(A_i)I_i + m_iA_i]}D）. The eigenvalue sign conditions also provide lower or upper bounds to the harmoni mean of the expected sojourn times of Markov chains. These, inequalities appear in the asymptoti growth rates of viral quasispecies, models for the evolution of dispersal in random environments, an the evolution of site-specific.mutation rates over the entire genome.