Hadamard inverses, square roots and products of almost semidefinite matrices Original Research Article

Let A = (aij) 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 image 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 image, 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 succeeds or equal to (1/eTB−1e)A.