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An Analogue of Birkhoff's Problem III for Infinite Markov Matrices1

Published online by Cambridge University Press:  20 November 2018

Choo-Whan Kim
Choo-Whan Kim*
Affiliation:
Simon Fraser University
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A celebrated theorem of Birkhoff ([1], [6]) states that the set of n × n doubly stochastic matrices is identical with the convex hull of the set of n × n permutation matrices. Birkhoff [2, p. 266] proposed the problem of extending his theorem to the set of infinite doubly stochastic matrices. This problem, which is often known as Birkhoffs Problem III, was solved by Isbell ([3], [4]), Rattray and Peck [7], Kendall [5] and Révész [8].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 5 , October 1969 , pp. 625 - 633
Copyright
Copyright © Canadian Mathematical Society 1969

Footnotes

1

This work was supported by the National Science Foundation under Grant GP-5270, Syracuse University.

References

1. Birkhoff, G., Tres observaciones sobre el algebra lineal. Rev. Univ. nac. Tucuman A. 5 (1946) 147–151.Google Scholar
2. Birkhoff, G., Lattice theory (revised edition). (New York, 1948).Google Scholar
3. Isbell, J.R., Birkhoff's Problem III. Proc. Amer. Math. Soc. 6 (1955) 217–218.Google Scholar
4. Isbell, J.R., Infinitely doubly stochastic matrices. Canad. Math. Bull. 5 (1962) 1–4.Google Scholar
5. Kendall, D.G., On infinite doubly stochastic matrices and Birkhoff's Problem 111. J. Lon. Math. Soc. 35 (1960) 81–84.Google Scholar
6. Marcus, M. and Mine, H., A survey of matrix theory and matrix inequalities. (Allyn and Bacon, Boston, 1964).Google Scholar
7. Rattray, B.A. and Peck, J. E. L., Infinite doubly stochasticmatrices. Trans. Roy. Soc. Canada III (3) 49 (1955) 55–57.Google Scholar
8. Revesz, P., A probabilistic solution of P roblem 111 of G. Birkhoff. Acta. Math. Acad. Sci. Hungar. 13 (1962) 187–198.Google Scholar