On the norm of the determinant of a complex positive definite matrix (Q2724942)

It is proved that (1) if \(A\) is an \(n\)-by-\(n\) positive definite complex matrix and \(B\) is an \(n\)-by-\(n\) positive semidefinite matrix with rank \(r(0<r<n)\), then \(|\det (A+B)|> |\det A|\), and (2) if NEWLINE\[NEWLINEA=\left( \begin{matrix} A_1 & A_2\\ A^*_2 & A_3 \end{matrix}\right)\text{ and }B=\left( \begin{matrix} B_1 & B_2\\ B^*_2 & B_3\end{matrix}\right)NEWLINE\]NEWLINE are \(n\)-by-\(n\) positive definite matrices, then NEWLINE\[NEWLINE\biggl|\det\bigl((A+B)/ (A_1+B_1)\bigr) \biggr |^2>\bigl |\det(A/A_1) \bigr|^2+ \bigl|\det (B/B_1)\bigr |^2,NEWLINE\]NEWLINE where \(A/A_1\) denotes \(A_3-A^*_2 A_1^{-1}A_2\), the Schur complement of \(A_1\) in \(A\) and similarly for \(B/B_1\) and \((A+B)/(A_1+B_1)\).