Hadamard inverses, square roots and products of almost semidefinite matrices

Abstract

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 $\text{A°}{}^{\mathrm{(-1)}}\text{= (}\text{1}\text{a}{}_{\mathrm{ij}}\text{)}$ 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 $\text{A°}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= (a}{}_{\mathrm{ij}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, 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 $\text{A ○ B ⪰ (1/}\text{e}{}^{\mathrm{<mtext>T</mtext>}}\text{B}{}^{\mathrm{-1}}\text{e}\text{)A}$.