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Article contents

Scalar matrix quadratic residues

Published online by Cambridge University Press:  26 February 2010

Gordon Pall
Affiliation:
Louisiana State University, California Institute of Technology.
Olga Taussky
Affiliation:
Louisiana State University, California Institute of Technology.

Extract

No systematic study seems to have been made of so natural a question as the analogue for matrices of quadratic residues. One generalization of x2 (x an integer) is X2 (X an integral matrix). Another is XX, where the prime means “transpose”. We study here the solvability for X of the congruence

where p is a prime, r ≥ 1; I (the identity matrix) and X are n-by-n; and a is an integer not divisible by p2.

Type
Research Article
Copyright
Copyright © University College London 1965

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References

1. Lagrange, J. L., Mém. Acad. Berlin, 1770.Google Scholar
2. Hilbert, D., Ges. Abhandlungen I (Springer, 1932, p. 164).CrossRefGoogle Scholar
3. Jones, B. W., The arithmetic theory of quadratic forms (Carus Mathematical Monograph, 1950).Google Scholar
4. Pall, G., “The arithmetical invariants of quadratic forms”, Bull. American Math. Soc, 51 (1945), 185197.CrossRefGoogle Scholar
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