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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(-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 AB.
MSC:
15A09Matrix inversion, generalized inverses
15A18Eigenvalues, singular values, and eigenvectors
15A45Miscellaneous inequalities involving matrices