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The Rank and Multiplicity Theorem for the Reduction of Quadratic Forms

Published online by Cambridge University Press:  03 November 2016

S.N. Afriat
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Extract

Direct and independent proof is given to the theorem, implicit in the possibility, and fundamental for the procedure of the simultaneous canonical reduction of a pair of real quadratic forms, one of which is positive definite. In this way a complete unification of the possibility and the procedure is effected.

Recently Ferrar (1) has considered this question of reduction by a method “free of difficult invariant factor arguments”, and Todd (2) the example of the orthogonal reduction of a single form in a direct and non-inductive fashion to remove, with reference to the traditional method for proving the possibility, “the aesthetic objection that it does not correspond to the practical method for obtaining the reduction”.

Type
Research Article
Information
The Mathematical Gazette , Volume 37 , Issue 319 , February 1953 , pp. 27 - 28
Copyright
Copyright © The Mathematical Association 1953

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References

(1) Ferrar, , “The Simultaneous Reduction of Two Real Quadratic Forms,” Quart. J. Math. Oxford, Ser. II, 186–92 (1947).Google Scholar
(2) Todd, , “A Note on Real Quadratic Forms,” Quart. J. Math. Oxford, Ser. II, 183–5 (1947).Google Scholar
(3) Aitken, , Determinants and Matrices (Oliver & Boyd, London, 1942), pp. 68, 89, 102.Google Scholar
(4) Turnbull, and Aitken, , Introduction to the Theory of Canonical Matrices (London and Glasgow, 1932), p. 91.Google Scholar