Positive definiteness of Hadamard exponentials and Hadamard inverses (Q6594794)

Assume that \(A\) is a Hermitian matrix. We denote by \((\pi(A), \zeta(A), \nu(A))\) its inertia, where \(\pi(A)\) is the number of positive eigenvalues of \(A\), \(\zeta(A)\) is the number of zero eigenvalues of \(A\), and \(\nu(A)\) is the number of negative eigenvalues of \(A\). We know that, for \(n\ge 2\), if \(A=[a_{ij}]\) is a positive semidefinite matrix, then the following claims are equivalent:\N\N(1) The Hadamard exponential \([e^{a_{ij}}]\) of \(A\) is positive definite;\N\N(2) No two columns of \(A\) are identical;\N\N(3) \(a_{ii}+a_{jj}>2a_{ij}\) for any distinct \(i, j\).\N\NIn this paper, the author gives an alternative proof of the above equivalences, along with an application to Hadamard inverses.