Vol. 19, No. 2, 1966

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ISSN: 0030-8730
Extreme copositive quadratic forms

Leonard Daniel Baumert

Vol. 19 (1966), No. 2, 197–204
Abstract

A real quadratic form Q = Q(x1,,xn) is called copositive if Q(x1,,xn) 0 whenever x1,,xn 0. If we associate each quadratic form Q = qijxixj qij = qji (i,j = 1,,n) with a point (q11,,qnn,√2-q12,,√2qn1,n) of Euclidean n(n + 1)2 space then the copositive forms constitute a closed covex cone in this space. We are concerned with the extreme points of this cone. That is, with those copositive quadratic forms Q for which Q = Q1 + Q2 (with Q1, Q2 copositive) implies Q1 = aQ, Q2 = (1 a)Q, 0 a 1. We show that if Q(x1,,xn), n 2, is an extreme copositive quadratic form then for any index i(1 i n), Q has a zero u with u1,un 0 where ui > 0. Further, using this fact, we establish an extension process whereby extreme forms in n variables can be used to construct extreme form in n variables, for all n> n.

Mathematical Subject Classification
Primary: 15.70
Secondary: 05.00
Milestones
Received: 5 April 1965
Published: 1 November 1966
Authors
Leonard Daniel Baumert