Quadratic forms positive definite on a linear manifold

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.