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Cauchy inequalities for the spectral radius of products of diagonal and nonnegative matrices

Author: Joel E. Cohen
Journal: Proc. Amer. Math. Soc. 142 (2014), 3665-3674
MSC (2010): Primary 15A42; Secondary 15B48, 15A16, 15A18, 26D15, 60K37
Published electronically: July 2, 2014
MathSciNet review: 3251708
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Abstract: Inequalities for convex functions on the lattice of partitions of a set partially ordered by refinement lead to multivariate generalizations of inequalities of Cauchy and Rogers-Hölder and to eigenvalue inequalities needed in the theory of population dynamics in Markovian environments: If $ A$ is an $ n\times n$ nonnegative matrix, $ n > 1$, $ D$ is an $ n\times n$ diagonal matrix with positive diagonal elements, $ r(\cdot )$ is the spectral radius of a square matrix, $ r(A)>0 $, and $ x \in [1,\infty )$, then $ r^{x-1}(A) r(D^xA) \geq r^x(DA) $. When $ A$ is irreducible and $ A^T A$ is irreducible and $ x>1$, then equality holds if and only if all elements of $ D$ are equal. Conversely, when $ x>1$ and $ r^{x-1}(A)r(D^xA)=r^x(DA)$ if and only if all elements of $ D$ are equal, then $ A$ is irreducible and $ A^T A$ is irreducible.

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Additional Information

Joel E. Cohen
Affiliation: Laboratory of Populations, The Rockefeller University and Columbia University, 1230 York Avenue, Box 20, New York, New York 10065

Received by editor(s): November 13, 2012
Received by editor(s) in revised form: November 14, 2012
Published electronically: July 2, 2014
Communicated by: Walter Craig
Article copyright: © Copyright 2014 Joel E. Cohen

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