Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-29T19:37:52.674Z Has data issue: false hasContentIssue false

Quadratic forms positive definite on a linear manifold

Published online by Cambridge University Press:  24 October 2008

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

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
Copyright
Copyright © Cambridge Philosophical Society 1952

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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