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 |(A^*MB)|²≤ (A^*MA)· (B^*MB), A,B∈ Cm× n, where Mm× m is hermitian positive definite. Here m and n are arbitrary. In case m≤ n, equality holds trivially. Equality holds when $m>n$ and rank(A)=rank(B)=n if and only if the columns of A and the columns of B span the same subspace of Cᵐ.