Finite criteria for conditional definiteness of quadratic forms

AbstractA real symmetric n × n matrix Q is A-conditionally positivesemidefinite, where A is a given m × n matrix, if x′Qx⩾0 whenever Ax⩾0, and is A-conditionally positive definite if strict inequality holds except when x=0. When A is the identity matrix these notions reduce to the well-studied notions of copositivity and strict copositivity respectively. This paper presents finite criteria, involving only the solution of sets of linear equations constructed from the matrices Q,A, for testing both types of conditional definiteness. These criteria generalize known facts about copositive matrices and, when Q is invertible and all row submatrices of A have maximal rank, can be very elegantly stated in terms of Schur complements of the matrix AQ-1A′