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The Term and Stochastic Ranks of a Matrix

Published online by Cambridge University Press:  20 November 2018

N. S. Mendelsohn  and
A. L. Dulmage
N. S. Mendelsohn
Affiliation:
University of Manitoba
A. L. Dulmage
Affiliation:
University of Manitoba
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The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.

The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 269 - 279
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Dulmage, A. L. and Mendelsohn, N. S., Some generalizations of the problem of distinct representatives Can. J. Math., 10 (1958), 230-41.Google Scholar
2. Dulmage, A. L. The convex hull of sub-permutation matrices, Proc. Amer. Math. Soc, 9 (1958), 253-4.Google Scholar
3. Dulmage, A. L. Coverings of bipartite graphs, Can. J. Math., 10 (1958), 517-34.Google Scholar
4. Ryser, H. J., Combinatorial properties of matrices of zeros and ones, Can. J. Math., 9 (1957), 371–7.Google Scholar
5. Ryser, H. J. The term rank of a matrix, Can. J. Math., 10 (1957), 57–65.Google Scholar