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Integral p-adic Normal Matrices Satisfying the Incidence Equation

Published online by Cambridge University Press:  20 November 2018

J. K. Goldhaber
J. K. Goldhaber*
Affiliation:
Washington University
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The problem of arranging v elements into v sets in such a way that every set contains exactly k distinct elements and that every pair of sets has exactly λ = k(k — l)/(v — 1) elements in common, where 0 < » < k < v, is equivalent to finding a normal integral v by v matrix A such that AT A = B, where B is the v by v matrix having k in every position on the main diagonal and λ in all other positions (10). Utilizing the fact that for the existence of a λ, k, v design it is necessary that I (the v by v identity matrix) represent B rationally, (2) and (3) have proved the non-existence of certain λ, k, v designs. Neither of the proofs utilize the fact that it is necessary that A be normal. However, Albert (1) for the projective plane case and Hall and Ryser (5) for the general design proved that if there exists a rational A such that ATA = B then there exists a normal rational matrix satisfying the same equation. Thus the requirement of normality does not exclude any λ, k, v which were not previously excluded.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 126 - 133
Copyright
Copyright © Canadian Mathematical Society 1960

References

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