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Scalar matrix quadratic residues

Published online by Cambridge University Press:  26 February 2010

Gordon Pall  and
Olga Taussky
Gordon Pall
Affiliation:
Louisiana State University, California Institute of Technology.
Olga Taussky
Affiliation:
Louisiana State University, California Institute of Technology.
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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 X′ X, 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
Information
Mathematika , Volume 12 , Issue 1 , June 1965 , pp. 94 - 96
Copyright
Copyright © University College London 1965

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References

1. Lagrange, J. L., Mém. Acad. Berlin, 1770.Google Scholar
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4. Pall, G., “The arithmetical invariants of quadratic forms”, Bull. American Math. Soc, 51 (1945), 185–197.CrossRefGoogle Scholar