Constructing copositive matrices from interior matrices

Abstract

Abstract. Let A be an n by n symmetric matrix with real entries. Using the l 1 -norm for vectors and letting S + 1 = {x ∈ R n |||x|| 1 = 1, x ≥ 0}, the matrix A is said to be interior if the quadratic form x T Ax achieves its minimum on S + 1 in the interior. Necessary and sufficient conditions are provided for a matrix to be interior. A copositive matrix is referred to as being exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. A method is provided for constructing exceptional copositive matrices by completing a partial copositive matrix that has certain specified overlapping copositive interior principal submatrices.

Sufficient conditions to be exceptional

“…Theorem 4 improves on Theorem 1, since the signs of the entries, including the diagonal entries, of A −1 are not restricted to being nonnegative. This may be seen from the examples of exceptional matrices from [11] and [12] following the theorem.…”

“…Another illustration of the same theorem is the 7-by-7 extension of the Horn matrix given in [12], which is the exceptional matrix A, along with A −1 given by …”

“…A simple observation is that if P is a positive semidefinite matrix and N is nonnegative, then A = P + N is a copositive matrix. It is well-known (see [7], [8], [10], [12]) that copositive matrices do not have to be of this form, an example of which is the 5-by-5 matrix H (from above) that we called the Horn matrix in [12]. In fact the Horn matrix is extreme [10], i.e.…”

“…it cannot be written nontrivially as the convex sum of two copositive matrices. In [12] we called copositive matrices exceptional if they are not the trivial sum of a positive semidefinite matrix and a nonnegative matrix. Otherwise, we call them non-exceptional.…”

“…An example of a matrix A to illustrate Theorem 4 is the Hoffman-Pereira matrix [11], as we called it in [12], which is copositive. This exceptional A along with its inverse is …”