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Some Properties of the Rank and Invariant Factors of Matrices*

Published online by Cambridge University Press:  20 November 2018

R.E. Gomory ,
A.J. Hoffman  and
N.C. Hsu
R.E. Gomory
Affiliation:
I. B. MYorktown HeightsN. Y.
A.J. Hoffman
Affiliation:
I. B. MYorktown HeightsN. Y.
N.C. Hsu
Affiliation:
I. B. MYorktown HeightsN. Y.
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Some years ago, G. Pall observed that the invariant factors of the incidence matrices of a certain pair of non-isomorphic projective planes of order 9 were different. With the aim of investigating such phenomena experimentally, we have constructed a code to calculate the invariant factors of rational integral matrices (actually, we compute the Smith's normal form of these matrices, as described, e.g., in MacDuffee [2; p. 41]), and this note is in the nature of a report on some preliminary experiments in the use of this code. In particular, we have computed the invariant factors of all (0, 1) matrices of order ≤ 8, with constant row and column sums, and these data are presented in the Appendix.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 1 , March 1964 , pp. 85 - 96
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Dulmage, A. L.and Mendelsohn, N. S., The term and stochastic ranks of a matrix. Canadian J. Math., vol. 11, (1959), pp. 269–279.Google Scholar
2. MacDuffee, C. C., The theory of matrices. Chelsea, New York (1946).Google Scholar
3. Ryser, H. J., Combinatorial properties of matrices of zeros and ones. Canadian J. Math., vol. 9, (1957), pp.371–377.Google Scholar