Eigenvalue inequalities for products of matrix exponentials |
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Authors: | Joel E Cohen Shmuel Friedland Tosio Kato Frank P Kelly |
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Institution: | Rockefeller University New York, New York, USA;Hebrew University Jerusalem, Israel;University of California Berkeley, California, USA;University of Cambridge Cambridge, England |
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Abstract: | Motivated by models from stochastic population biology and statistical mechanics, we proved new inequalities of the form , where A and B are n × n complex matrices, 1<n<∞, and ? 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 ? is the spectral radius, then (1) holds; if in addition A is irreducible and B has at least two different diagonal elements, then the inequality (1) 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. |
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