Extreme Copositive Quadratic Forms

Citation

Baumert, Leonard Daniel (1965) Extreme Copositive Quadratic Forms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/TQKA-PF96. https://resolver.caltech.edu/CaltechETD:etd-01132003-105545

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A real quadratic form [...] is called copositive if [...] whenever [...]. If we associate each quadratic form [...] with a point [...] of Euclidean [...] space, then the copositive forms constitute a closed convex cone in this space. We are concerned with the extreme points of this cone. That is, with those copositive quadratic forms Q for which [...] implies [...]. We show that (1) If [...] is an extreme copositive quadratic form then for any index pair [...] has a zero [...] with [...]. (2) If [...] is an extreme copositive quadratic form in [...] variables [...] then replacing [...] in [...] yields a new copositive form [...] which is also extreme. (3) If [...] is an extreme copositive quadratic form then either (i) Q is positive semi-definite, or (ii) Q is related to an extreme form discovered by A. Horn, or (iii) Q possesses exactly five zeros having non-negative components. In this later case the zeros can be assumed to be [...] and [...] where [...].

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