Hadamard inverses, square roots and products of almost semidefinite matrices (Q1300864)

Hadamard products, inverses, and square roots of \(n\times n\) symmetric matrices \(A,B\) with all positive entries, having just one positive eigenvalue, are studied. The result by \textit{R. B. Bapat} [Proc. Am. Math. Soc. 102, No. 3, 467-472 (1988; Zbl 0647.60019)] on positive semidefiniteness of the Hadamard inverses \(A^{0(-1)}\) is extended. It is shown that if \(A\) is invertible then \(A^{0(-1)}\) is positive definite. Necessary and sufficient conditions are given on the invertibility of \(A^{0(-1)}\). The Hadamard square root has just one positive eigenvalue and is invertible if \(A\) is a symmetric matrix, with all diagonal entries zero. An inequality is derived for \(A\circ B\).