Some determinantal inequalities for Hadamard product of matrices (Q1399234)

The author proves the following Theorem: Let \(A=(a_{ij})\) and \(B=(b_{ij})\) be \(M\)-matrices or positive definite real symmetric matrices of order \(n.\) Then \[ \det (A\circ B) \geq \det (AB)\prod\limits_{k=2}^n\left( \frac{a_{kk}\det A_{k-1}}{\det A_k}+\frac{b_{kk}\det B_{k-1}}{\det B_k}-1\right),\tag{1} \] where \(A\circ B \) is the Hadamard product of \(A\) and \(B\) and \(A_k\) and \(B_k\) \((k=1,2,\dots,n)\) are the \(k\times k\) leading principal submatrices of \(A \) and \(B\). Other inequalities for Hadamard product are obtained.