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

2024/03/05 by Avram Sidi, Sidi, Avram
Mathematics · #15 #FOS: Mathematics #Functional Equations Stability Results #General Mathematics (math.GM) #Mathematical Inequalities and Applications #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2403.19691

openalex publication_date 2024/03/05 · openalex created_date 2024/04/03 · openalex updated_date 2026/07/28

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≤ det(A^*MA)⋅ det(B^*MB), A,B∈ ℂ^{m× n}, where M∈ℂ^{m× 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 ℂ^m.

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