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Quadratic forms positive definite on a linear manifold

Published online by Cambridge University Press:  24 October 2008

L. S. Goddard
L. S. Goddard
Affiliation:
Department of MathematicsKing's CollegeAberdeen
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Extract

1. In a recent paper(1), Afriat has given necessary and sufficient conditions for a real quadratic form to be positive definite on a linear manifold, in terms of the dual Grassmannian coordinates of the manifold. Considerable matrix manipulations were used in Afriat's method, but most of these may be avoided by the method of the present paper, which depends on some well-known properties of the Grassmannian coordinates. We first show that the conditions may be expressed as a set of inequalities which are quadratic in the Grassmannian coordinates of the manifold. Then, by a standard theorem, these may be transformed into Afriat's conditions on the dual coordinates.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 1 , January 1952 , pp. 70 - 71
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Afriat, S. N.The quadratic form positive definite on a linear manifold. Proc. Gamb. phil. Soc. 47 (1951), 1.Google Scholar
(2)Hodge, W. V. D. and Pedoe, D.Methods of algebraic geometry (Cambridge, 1947).Google Scholar