A conjugate decomposition of the Euclidean space

Given a closed convex cone K in the n-dimensional real Euclidean space R(n) and an nxn real matrix A that is positive definite on K, we show that each vector in R(n) can be decomposed into a component that lies in K and another that lies in the conjugate cone induced by A and such that the two vectors are conjugate to each other with respect to A + A(T). As a consequence of this decomposition we establish the following characterization of positive definite matrices: An nxn real matrix A is positive definite if and only if it is positive definite on some closed convex cone K in R(n) and (A + A(T))(-1) exists and is positive semidefinite on the polar cone K(0). If K is a subspace of R(n), then K(0) is its orthogonal complement K[unk]. Other applications include local duality results for nonlinear programs and other characterizations of positive definite and semidefinite matrices.