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A short proof of the Farahat-Mirsky refinement of Birkhoff's theorem on doubly-stochastic matrices

Published online by Cambridge University Press:  24 October 2008

J. M. Hammersley
J. M. Hammersley
Affiliation:
Trinity CollegeOxford
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Extract

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.

Type
Research Notes
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 3 , July 1961 , pp. 681
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Farahat, H. K. and Mirsky, L.Permutation endomorphisms and a refinement of a theorem of Birkhoff. Proc. Camb. Phil. Soc. 56 (1960), 322–8.Google Scholar
(2)Hammersley, J. M. and Mauldon, J. G.General principles of antithetic variates. Proc. Camb. Phil. Soc. 52 (1956), 476–81.Google Scholar
(3)Handscomb, D. C.Proof of the antithetic-varjates theorem for n > 2. Proc. Camb. Phil. Soc. 54 (1958), 300–1.Google Scholar