Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T12:44:14.214Z Has data issue: false hasContentIssue false
  • English
  • Français

Contractive inhomogeneous products of non-negative matrices

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. Hajnal (5) showed that under wide conditions a sequence of products H(1, q) = Mq…M1q = 1,2,…, of square non-negative matrices Mq approaches a sequence of positive matrices of rank 1. We call a product H(1,q) inhomogeneous if its factors M1,…, Mq are not necessarily all equal to one another. When the matrices Mq are members of an ‘ergodic set’, and x and y are positive vectors, the projective distance d(H(1,q)x, H(1,q)y) decays at least exponentially fast as q increases. An important condition on an ergodic set is that any product of g members from the set be a matrix in which all elements are (strictly) positive, where g is some fixed positive integer.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 2 , September 1979 , pp. 351 - 364
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)Birkhoff, G. and Varga, R. S.Reactor criticality and nonnegative matrices. J. Soc. Indust. App. Math. 6 (1958), 354–377.CrossRefGoogle Scholar
(2)Brauer, A. On the characteristic roots of non-negative matrices. In Recent advances in matrix theory, ed. Schneider, H. (Madison: University of Wisconsin Press, 1964), pp. 3–38.Google Scholar
(3)Bushell, P.Hilbert's metric and positive contraction mappings in a Banach space. Arch. Rational Mech. Anal. 52 (1973), 330–338.CrossRefGoogle Scholar
(4)Gantmacher, F. R.The theory of matrices, vol. 2 (New York, Chelsea, 1960).Google Scholar
(5)Hajnal, J.On products of non-negative matrices. Math. Proc. Cambridge Philos. Soc. 79 (1976), 521–530.CrossRefGoogle Scholar
(6)Kaijser, T.A limit theorem for partially observed Markov chains. Ann. Probability 3 (1975), 667–696.CrossRefGoogle Scholar
(7)Kablin, S.Equilibrium behaviour of population genetic models with non-random mating (Gordon and Breach, 1968).Google Scholar
(8)Lange, K.On Cohen's stochastic generalization of the strong ergodic theorem of demography. J. Appl. Probability (in the Press).Google Scholar
(9)Loève, M.Probability theory, 3rd ed. (New York, Van Nostrand, 1963).Google Scholar
(10)Seneta, E.Non-negative matrices (George Allen and Unwin, 1973).Google Scholar
(11)Stone, R.A Markovian education model and other examples linking social behaviour to the economy. J. Roy. Statist. Soc. Section A 135 (1972), 511–532.CrossRefGoogle Scholar
(12)Till, J. E., McCulloch, E. A. and Siminovitch, L.A stochastic model of stem cell proliferation, based on the growth of spleen colony-forming cells. Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 29–36.CrossRefGoogle ScholarPubMed