A copositivity probe (Q5954132)

A real symmetric matrix \(A\) of order \(p\) is said to be copositive if \(x^TAx \geq 0\) for \(x \geq 0\) (strictly copositive if it is copositive and equality holds only for \(x=0\)). The aim of the paper is to find whether a real symmetric matrix \(A\) is equal to the sum of a positive semidefinite matrix \(S\) and a nonnegative matrix \(P\). Every such matrix \(A\) is copositive and it is strictly copositive if \(S\) is positive definite. NEWLINENEWLINENEWLINEA very simple algorithm based on the perturbation theory of eigenvalues and eigenvectors in quantum mechanics is used here. It is shown that if a tested matrix \(A\) is not copositive then in many such cases the algorithm produces a proof that \(A\) is not copositive by producing a positive vector \(v\) such that \(v^TAv < 0\). This byproduct increases the value of the procedure. If the procedure succeeds, one has a proof of copositivity and if it fails and produces the mentioned vector \(v\), one has a proof of noncopositivity.