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Hadamard inverses, square roots and products of almost semidefinite matrices. (English) Zbl 0933.15006
Hadamard products, inverses, and square roots of $n×n$ symmetric matrices $A,B$ with all positive entries, having just one positive eigenvalue, are studied. The result by R. B. Bapat [Proc. Am. Math. Soc. 102, No. 3, 467-472 (1988; Zbl 0647.60019)] on positive semidefiniteness of the Hadamard inverses ${A}^{0\left(-1\right)}$ is extended. It is shown that if $A$ is invertible then ${A}^{0\left(-1\right)}$ is positive definite. Necessary and sufficient conditions are given on the invertibility of ${A}^{0\left(-1\right)}$. 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 $A\circ B$.
##### MSC:
 15A09 Matrix inversion, generalized inverses 15A18 Eigenvalues, singular values, and eigenvectors 15A45 Miscellaneous inequalities involving matrices