Analogue of the Cauchy-Schwarz inequality for determinants: a simple proof

In this note, we present a simple proof of an analogue of the Cauchy-Schwarz inequality relevant to products of determinants. Specifically, we show that $$ |\det(A^*MB)|^2\leq \det(A^*MA)\cdot \det(B^*MB),\quad A,B\in \mathbb{C}^{m\times n},$$ where $M\in\mathbb{C}^{m\times m}$ is hermitian positive definite. Here $m$ and $n$ are arbitrary. In case $m\leq n$, equality holds trivially. Equality holds when $m>n$ and $\text{rank}(A)=\text{rank}(B)=n$ if and only if the columns of $A$ and the columns of $B$ span the same subspace of $\mathbb{C}^m$.
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