A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices

TL;DR

This paper provides a concise proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices, using only basic properties of determinants and real eigenvalues.

Contribution

It offers a remarkably short and straightforward proof of a classical eigenvalue interlacing theorem for Hermitian matrices.

Findings

Proof uses only linearity of determinant and real eigenvalues

The proof is only two sentences long

Clarifies the fundamental nature of eigenvalue interlacing

Abstract

Cauchy's interlace theorem states that the characteristic polynomial of a symmetric matrix is interlaced by the characteristic polynomial of any principle submatrix. We prove this in two sentences using only the linearity of the determinant, and the fact that all eigenvalues of a symmetric matrix are real.