Finite Criteria for Conditional Definiteness of Quadratic Forms

摘要

Finite criteria, involving the solution of sets of linear equations constructed from the matrices Q,A (Q symmetric and n x n, A is m x n), for testing two types of conditional definiteness are presented. The real symmetric n x n matrix Q is A-conditionally positive semidefinite, where A is a given m x n real matrix, if x prime Q x or = O whenever Ax or = O, and is A-conditionally positive definite if strict inequality holds except when x = O. When A is the identity matrix these notions reduce to the well studied notions of copositivity and strict copositivity respectively. The criteria developed generalize known facts about copositive matrices, and, when Q is invertible and all rowsubmatrices of A have maximal rank, can be very elegantly stated in terms of Schur complements of the matrix AQ (-1) A prime.