Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-07-30T04:25:19.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Lower bounds for the Perron root of a non-negative irreducible matrix

Published online by Cambridge University Press:  24 October 2008

Emeric Deutsch
Emeric Deutsch
Affiliation:
Polytechnic Institute of New York
Get access Rights & Permissions [Opens in a new window]

Extract

We derive a family of lower bounds for the Perron root of a non-negative irreducible matrix. These lower bounds are better than certain lower bounds of the Rayleigh quotient type, also derived in this paper. For the particular case of a symmetric non-negative irreducible matrix, our lower bound is always better than a corresponding Rayleigh quotient and, as shown in example 4, it can be infinitely better.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 1 , July 1982 , pp. 49 - 54
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Berman, A. and Piemmons, R. J.Non-negative matrices in the mathematical sciences (Academic Press, New York, 1979).Google Scholar
(2)Brauer, A. and Gentry, I. C.Bounds for the greatest characteristic root of an irreducible non-negative matrix: II. Linear Algebra and Appl. 13 (1976), 109–114.CrossRefGoogle Scholar
(3)Cohen, J. E.Random evolutions and the spectral radius of a non-negative matrix. Math. Proc. Cambridge Philos. Soc. 86 (1979), 345–350.CrossRefGoogle Scholar
(4)Deutsch, E.Bounds for the Perron root of a non-negative irreducible partitioned matrix. Pacific J. Math. 92 (1981), 49–56.CrossRefGoogle Scholar
(5)Deutsch, E. and Wielandt, H. Nested bounds for the Perron root of a non-negative matrix. Linear Algebra and Appl. (to appear.)Google Scholar
(6)Friedland, S. and Karlin, S.Some inequalities for the spectral radius of non-negative matrices and applications. Duke Math. J. 42 (1975), 459–490.CrossRefGoogle Scholar
(7)Gantmacher, F. R.The theory of matrices, vol. 2 (Chelsea, New York, 1959).Google Scholar
(8)Gregory, R. T. and Karney, D. L.A Collection of matrices for testing computational algorithms (Wiley-Interscience, New York, 1969).Google Scholar
(9)Marcus, M. and Ming, H.A survey of matrix theory and matrix inequalities (Allyn and Bacon, Boston, 1964).Google Scholar
(10)Merikoski, J. K.On a lower bound for the Perron eigenvalue. BIT 19 (1979), 39–42.CrossRefGoogle Scholar
(11)Rothblum, U. G. Private communication.Google Scholar
(12)Varga, R. S.Matrix iterative analysis (Prentice-Hall, Englewood Cliffs, 1962).Google Scholar