Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] L. Altenberg, The evolution of dispersal in random environments and the principle of partial control, Ecological Monographs 82 (3) (2012) 297-333.
  • [2] -, A sharpened condition for strict log-convexity of the spectral radius via the bipartite graph, Linear Algebra Appl. 438 (2013), no. 9, 3702-3718. MR 3028608
  • [3] Joel E. Cohen, Stochastic population dynamics in a Markovian environment implies
    Taylor's power law of fluctuation scaling
    , Theoret. Population Biol. 93 (2014), 30-37. DOI 10.1016/j.tpb.2014.01.001
  • [4] Joel E. Cohen, Shmuel Friedland, Tosio Kato, and Frank P. Kelly, Eigenvalue inequalities for products of matrix exponentials, Linear Algebra Appl. 45 (1982), 55-95. MR 660979 (84h:15020),
  • [5] D. R. Cox, Renewal theory, Methuen & Co. Ltd., London, 1962. MR 0153061 (27 #3030)
  • [6] S. L. Feld, Why your friends have more friends than you do, Amer. J. Sociology 96 (6) (1991), 1464-1477.
  • [7] S. L. Feld and B. Grofman, Variation in class size, the class size paradox, and some consequences for students, Research in Higher Education 6 (3) (1977), 215-222.
  • [8] S. L. Feld and B. Grofman, Puzzles and paradoxes involving averages: an intuitive approach, Collective Decision Making: Views from Social Choice and Game Theory, Ad van Deemen and Agnieszka Rusinowska, eds., Springer Verlag, Berlin, 2010, pp. 137-150. DOI 10.1007/978-3-642-02865-6_10
  • [9] Shmuel Friedland, Convex spectral functions, Linear and Multilinear Algebra 9 (1980/81), no. 4, 299-316. MR 611264 (82d:15014),
  • [10] S. Karlin, Classifications of selection-migration structures and conditions for a protected polymorphism, Evolutionary Biology, 14, M. K. Hecht, B. Wallace, and G. T. Prance, eds., Plenum Publishing Corporation, New York, 1982, pp. 61-204.
  • [11] J. F. C. Kingman, A convexity property of positive matrices, Quart. J. Math. Oxford Ser. (2) 12 (1961), 283-284. MR 0138632 (25 #2075)
  • [12] Henryk Minc, Nonnegative matrices, A Wiley-Interscience Publication, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., New York, 1988. MR 932967 (89i:15001)
  • [13] J. Michael Steele, The Cauchy-Schwarz master class, An introduction to the art of mathematical inequalities. MAA Problem Books Series, Mathematical Association of America, Washington, DC, 2004. MR 2062704 (2005a:26035)
  • [14] S. D. Tuljapurkar, Population dynamics in variable environments. II. Correlated environments, sensitivity analysis and dynamics, Theoret. Population Biol. 21 (1982), no. 1, 114-140. MR 662525 (84g:92037a),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 15A42, 15B48, 15A16, 15A18, 26D15, 60K37

Retrieve articles in all journals with MSC (2010): 15A42, 15B48, 15A16, 15A18, 26D15, 60K37

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

American Mathematical Society