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On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, I: Square Matrices

Published online by Cambridge University Press:  20 November 2018

Kulendra N. Majindar
Kulendra N. Majindar*
Affiliation:
Loyola College, Montreal
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A few years ago, in a short paper (4) Ryser introduced an interesting topic in number theory, viz. the connection between integer matrices (i.e., matrices having only integers as their elements) satisfying certain conditions and 0-1 matrices (i.e., matrices that have no element different from 0 and 1). In this series of papers we shall pursue this topic further.

To make the statements of our theorems short we introduce some terminology. We need the definitions of certain 0-1 matrices related to a few well-known combinatorial configurations. By an incidence matrix of a balanced incomplete block (b.i.b. for conciseness) design we mean a 0-1 matrix with v rows and b columns, such that the sum of the elements in each column of A is k, k < v, and the scalar product of any two row vectors of A is λ, λ ≠ 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 1 - 5
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Bruck, R. H. and Ryser, H. J., The non-existence of certain finite projective planes, Can. J. Math., 1 (1949), 88–93.Google Scholar
2. Chowla, S. and Ryser, H. J., Combinatorial problems, Can. J. Math., 2 (1950), 93–99.Google Scholar
3. Majumdar, Kulendra N., On some theorems in combinatorics relating to incomplete block designs, Ann. Math. Statist., 24 (1953), 377–389.Google Scholar
4. Ryser, H. J., Matrices with integer elements in combinatorial investigations, Amer. J. Math., 74 (1952), 769–773.Google Scholar
5. Schutzenberger, M. P., A non-existence theorem for an infinite family of block designs, Ann. Eugenics, 14 (1939), 286–287.Google Scholar