Hadamard inverses, square roots and products of almost semidefinite matrices

Let A = (a(ij)) be an n x n symmetric matrix with all positive entries and just one positive eigenvalue. Bapat proved then that the Hadamard inverse o f A, given by A degrees((-1)) = (1/a(ij)) is positive semidefinite. We show that if moreover A is invertible then A degrees((-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 squ are root of A, given by A degrees(1/2) = (a(ij)(1/2)), has just one positiv e 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 circle B greater than or equal to (1/e(T)B(-1)e)A. (C) 1999 Elsevier Scien ce Inc. All rights reserved.