Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-07T02:04:03.239Z Has data issue: false hasContentIssue false
  • English
  • Français

Random evolutions and the spectral radius of a non-negative matrix

Published online by Cambridge University Press:  24 October 2008

Joel E. Cohen
Joel E. Cohen
Affiliation:
The Rockefeller University, New York
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction and summary. This paper offers yet another example of what probability theory can do for analysis. Using a Feynman-Kac formula derived in the theory of random evolutions (5), we find an expression (1) for the spectral radius r(A) of a finite square non-negative matrix A. This expression makes it very easy to study how r(A) behaves as a function of the diagonal elements of A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 2 , September 1979 , pp. 345 - 350
Copyright
Copyright © Cambridge Philosophical Society 1979

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)Chung, K. L.Markov chains with stationary transition probabilities (New York, Springer-Verlag, 1967).Google Scholar
(2)Cohen, J. E.Derivatives of the spectral radius as a function of non-negative matrix elements. Math. Proc. Cambridge Philos. Soc. 83 (1978), 183–190.CrossRefGoogle Scholar
(3)Donsker, M. D. and Varadhan, S. R. S.On a variational formula for the principal eigen-value for operators with maximum principle. Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 780–783.CrossRefGoogle Scholar
(4)Gantmacher, F. R.The theory of matrices, vol. 2 (New York, Chelsea, 1960).Google Scholar
(5)Griego, R. and Hersh, R.Theory of random evolutions with applications to partial differential equations. Trans. Amer. Math. Soc. 156 (1971), 405–418.CrossRefGoogle Scholar
(6)Hardy, G. H., Littlewood, J. E. and Pólya, G.Inequalities, 2nd ed. (Cambridge University Press, 1952).Google Scholar
(7)Kac, M. On some connections between probability theory and differential and integral equations. 2nd Berkeley Symp. Math. Stat. Probab., ed. Neyman, J. (Los Angeles, University of California Press, 1951), pp. 189–215.Google Scholar
(8)Kato, T.Perturbation theory for linear operators, 2nd ed. (New York, Springer-Verlag, 1976).Google Scholar
(9)Lancaster, P.Theory of matrices (New York, Academic Press, 1969; reproduced by the author, Calgary, 1977).Google Scholar
(10)Loève, M.Probability theory, 3rd ed. (Princeton, Van Nostrand, 1963).Google Scholar
(11)Marcus, M. and Minc, H.A survey of matrix theory and matrix inequalities (Boston, Allyn and Bacon, 1964).Google Scholar
(12)Muir, T.A treatise on the theory of determinants, rev. Metzler, W. H. (privately published, Albany, New York; reprinted by Dover, New York).Google Scholar
(13)Pólya, G. and Szegö, G.Problems and theorems in analysis, vol. 1, 4th ed. (New York, Springer-Verlag, 1972).Google Scholar
(14)Seneta, E.Non-negative matrices; an introduction to theory and applications (London, George Allen and Unwin, 1973).Google Scholar