Hadamard Inverses, Square Roots and Products of Almost Semidefinite Matrices

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 A = ( 1 aij ) is positive semidefinite. We show that if moreover A is invertible then A 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 1 eT Be A.