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The game-theoretic value and the spectral radius of a nonnegative matrix

Authors: Joel E. Cohen and Shmuel Friedland
Journal: Proc. Amer. Math. Soc. 93 (1985), 205-211
MSC: Primary 15A42; Secondary 15A48, 90D05
MathSciNet review: 770520
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Abstract: We relate some minimax functions of matrices to some spectral functions of matrices. If $ A$ is a nonnegative $ n \times n$ matrix, $ \upsilon (A)$ is the game-theoretic value of $ A$, and $ \rho (A)$ is the spectral radius of $ A$, then $ \upsilon (A) \leq \rho (A)$. Necessary and sufficient conditions for $ \upsilon (A) = \rho (A)$ are given. It follows that if $ A$ is nonnegative and irreducible and $ n > 1$, then $ \upsilon (A) < \rho (A)$. Also, if, for a real matrix $ A$ and a positive matrix $ B$, $ \upsilon (A,B) = {\sup _X}{\inf _Y}{X^T}AY/{X^T}BY$ over probability vectors $ X$ and $ Y$, then for nonnegative, nonsingular $ A$ and positive $ B$, $ \rho (AB) = {[\upsilon ({A^{ - 1}},B)]^{ - 1}}$.

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Keywords: Eigenvalue inequality, Perron-Frobenius root, minimax, inverse nonnegative matrix, essentially nonnegative matrix, zero-sum two-person game, Jentzsch's theorem
Article copyright: © Copyright 1985 American Mathematical Society

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