Constructing copositive matrices from interior matrices

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Charles R. Johnson
Robert Reams


Let A be an n by n symmetric matrix with real entries. Using the l1-norm for vectors and letting S1+= {x ∈ ℝn|||x||1 = 1, x ≥ 0}, the matrix A is said to be interior if the quadratic form xTAx achieves its minimum on S1+ 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.

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