Let A = (a ij) be an n × n symmetric matrix with all positive entries and just one positive eigenvalue. Bapat proved then that the Hadamard inverse of A, given by A •(−1) = (1 a ij) is positive semidefinite. We show that if moreover A is invertible then A •(−1) is positive definite. We use this result to obtain a simple proof that with the same hypotheses on A, except that all the diagonal entries of A are zero, the Hadamard square root of A, given by A • 1 2 = (a 1 2 ij), has just one positive eigenvalue and is invertible. Finally, we show that if A is any positive semidefinite matrix and B is almost positive definite and invertible then A • B

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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.

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