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)

 
 

 

Convexity of the dominant eigenvalue of an essentially nonnegative matrix


Author: Joel E. Cohen
Journal: Proc. Amer. Math. Soc. 81 (1981), 657-658
MSC: Primary 15A42; Secondary 15A48, 92A15
DOI: https://doi.org/10.1090/S0002-9939-1981-0601750-2
MathSciNet review: 601750
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The dominant eigenvalue of a real $ n \times n$ matrix $ A$ with nonnegative elements off the main diagonal is a convex function of the diagonal of $ A$. We give a short proof using Trotter's product formula and a theorem on log-convexity due to Kingman.


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

  • [1] J. E. Cohen, Derivatives of the spectral radius as a function of non-negative matrix elements, Math. Proc. Cambridge Philos. Soc. 83 (1978), 183-190. MR 0466178 (57:6059)
  • [2] -, Random evolutions and the spectral radius of a non-negative matrix, Math. Proc. Cambridge Philos. Soc. 86 (1979), 345-350. MR 538756 (81c:15023)
  • [3] S. Friedland, Convex spectral functions, Linear and Multilinear Algebra (to appear). MR 611264 (82d:15014)
  • [4] J. F. C. Kingman, A convexity property of positive matrices, Quart. J. Math. Oxford Ser. (2) 12 (1961), 283-284. MR 0138632 (25:2075)
  • [5] E. Seneta, Non-negative matrices: an introduction to theory and applications, George Allen and Unwin, London, 1973. MR 0389944 (52:10773)
  • [6] H. F. Trotter, On the product of semigroups of operators, Proc. Amer. Math. Soc. 10 (1959), 545-551. MR 0108732 (21:7446)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A42, 15A48, 92A15

Retrieve articles in all journals with MSC: 15A42, 15A48, 92A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0601750-2
Keywords: Perron-Frobenius root, convexity, log-convexity, Trotter product formula, spectral radius
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society