Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-18T18:55:01.549Z Has data issue: false hasContentIssue false
  • English
  • Français

Notes on the Birkhoff Algorithm for Doubly Stochastic Matrices

Published online by Cambridge University Press:  20 November 2018

Richard A. Brualdi
Richard A. Brualdi*
Affiliation:
Department of Mathematics University of Wisconsin, Madison, WI 53706, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this note is to tie together some results concerning doubly stochastic matrices and their representations as convex combinations of permutation matrices.

Keywords

15A5105C5052A25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 2 , 01 June 1982 , pp. 191 - 199
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Birkhoff, G., Tres observaciones sobre el algebra lineal, Univ. Nac. Tucuman Rev. Ser. A, 5 (1946), 147-150.Google Scholar
2. Brualdi, R. A., The diagonal hypergraph of a matrix (bipartite graph), Discrete Math. 27 (1979), 127-147.Google Scholar
3. Brualdi, R. A. and Gibson, P. M., Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function, J. Comb. Theory Ser. A. 22 (1977), 194-230.Google Scholar
4. Brualdi, R. A. and Hedrick, M. B., A unified treatment of nearly reducible and nearly decomposable matrices, Lin. Alg. and its Applies. 24 (1979), 51-73.Google Scholar
5. Grunbaum, B., Convex Poly topes, Interscience, New York, 1967.Google Scholar
6. Hartfiel, D. J., A simplified form for nearly reducible and nearly decomposable matrices, Proc. Amer. Math. Soc. 24 (1970), 388-393.Google Scholar
7. Johnson, D. N., Dulmage, A. L., and Mendelsohn, N. S., On an algorithm of G. Birkhoff concerning doubly stochastic matrices, Canad. Math. Bull. 3 (1960), 237-242.Google Scholar
8. Marcus, M. and Ree, R., Diagonals of doubly stochastic matrices, Quart. J. Math. Oxford, Ser. (2), 10 (1959), 296-302.Google Scholar
9. Nishi, A., An elementary proof of Johnson-Dulmage-Mendelsohn's refinement of Birkhoff's theorem on doubly stochastic matrices, Canad. Math. Bull. 22 (1979), 81-86.Google Scholar
10. Ryser, H. J., Combinatorial Mathematics, Cams Math. Monograph No. 14, Math. Assoc, of America, Washington, D.C., 1963.Google Scholar
11. Schneider, H., The Birkhoff-Egervary-König theorem for matrices over lattice-ordered groups, Acta Math. Acad. Sci. Hungar. 30 (1977), 91-94.Google Scholar
12. Witte, F., Doubly stochastic matrices and sequential data association, Part I, Information linkage between applied mathematics and industry, 641-646, Academic Press, New York, 1979.Google Scholar
13. Witte, F. and Lucas, D., Probabilistic tracking in a multitarget environment, I.E.E.E., 1978, 1212-1219.Google Scholar