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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

1. Albert, A.A., Rational normal matrices satisfying the incidence equation, Proc. Amer. Math. Soc, 4 (1953), 554–9.Google Scholar
2. Bruck, R.H. and Ryser, H.J., The nonexistence of certain finite projective planes, Can. J. Math., 1 (1949), 88–93.Google Scholar
3. Chowla, S. and Ryser, H. J., Combinational problems, Can. J. Math., 2 (1950), 93–9.Google Scholar
4. Eichler, M., Quadratische formen und orthogonale gruppen (Berlin, 1952).Google Scholar
5. Hall, Marshall and Ryser, H.J., Normal completions of incidence matrices, Amer. J. Math., 76 (1954), 581–9.Google Scholar
6. Jones, B.W., A canonical quadratic form for the ring of 2-adic integers, Duke Math. J., 11 (1944), 715–27.Google Scholar
7. Jones, B.W., The arithmetic theory of quadratic forms, Carus Math. Monographs, 10 (1950).Google Scholar
8. Magnus, W., Ueber die Anzahl der in einem Geschlecht enthaltenen Klassen von positiv definiten quadratischen Formen, Math. Ann. 114 (1937), 465-75.Google Scholar
9. Mayer, A., Zurich naturf. Ges., 36 (1891), 241.Google Scholar
10. Ryser, H.J., Matrices with integer elements in combinational investigations, Amer. J. Math., 74 (1952), 769–73.Google Scholar
11. Siegel, C.L., Equivalence of quadratic forms, Amer. J. Math. 68 (1941), 658-80.Google Scholar