Determinantal inequalities for block triangular matrices

Abstract

Let $T=\left[\begin{array}{cc}X& Y\\ 0& Z\end{array}\right]$ be an $n$-square matrix, where $X,Z$ are $r$-square and $(n-r)$-square, respectively. Among other determinantal inequalities, it is proved $\det (I_n+T^{\ast }T)\ge \det (I_r+X^{\ast }X)\cdot \det (I_{n-r}+Z^{\ast }Z)$ with equality holds if and only if $Y=0$.

8 pages