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An inequality on permanents of Hadamard products. (English) Zbl 0969.15006

The author conjectures the following inequality: Let \(A\) and \(B\) be two positive semidefinite complex \(n\times n\) matrices \(A\) and \(B\). Further let \(a\) (\(b\)) be the product of all elements in the main diagonal of \(A\) (\(B\)). Then the permanent of the Hadamard product \(A\circ B\) is less or equal the maximum of \(\{b\text{ per }A, a\text{ per }B\}\). It is shown that this conjecture is true for \(n=2,3\). Also, the result holds in general if, and only if, it holds for all correlation matrices, i.e., positive definite matrices with all diagonal entries equal to \(1\).

MSC:

15A45 Miscellaneous inequalities involving matrices
15A15 Determinants, permanents, traces, other special matrix functions
15B57 Hermitian, skew-Hermitian, and related matrices
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