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

Avram Sidi

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

Avram Sidi

Abstract

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 $\mathbb{C}^m$.

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

Abstract

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

where

is hermitian positive definite. Here

and

are arbitrary. In case

, equality holds trivially. Equality holds when

and

if and only if the columns of

and the columns of

span the same subspace of

Paper Structure (3 theorems, 22 equations)

This paper contains 3 theorems, 22 equations.

Key Result

Lemma 1

Let $U$ and $V$ be two rectangular unitary matrices in $\hbox{$\mathbb{C}$}^{m\times n}$ with $m>n$, in the sense Then Equality holds if and only if the columns of $U$ and the columns of $V$ span the same subspace of $\hbox{$\mathbb{C}$}^m$.

Theorems & Definitions (5)

Lemma 1
proof
Theorem 2
proof
Theorem 3