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

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Abstract | References | Similar Articles | Additional Information

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

**[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)**

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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