Eigenvalue inequalities for products of matrix exponentials

AbstractMotivated by models from stochastic population biology and statistical mechanics, we proved new inequalities of the form (∗) ϕ(eAeB)⩾ϕ(eA+B), where A and B are n × n complex matrices, 1<n<∞, and ϕ is a real-valued continuous function of the eigenvalues of its matrix argument. For example, if A is essentially nonnegative, B is diagonal real, and ϕ is the spectral radius, then (∗) holds; if in addition A is irreducible and B has at least two different diagonal elements, then the inequality (∗) is strict. The proof uses Kingman's theorem on the log-convexity of the spectral radius, Lie's product formula, and perturbation theory. We conclude with conjectures