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

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.

arXiv comment1 page

Cited by in corpus (15)

• Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
• Survey article: Seventy years of Salem numbers
• Cyclotomic matrices over the Eisenstein and Gaussian integers
• On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix
• Counting substructures and eigenvalues I: triangles
• Cyclotomic Matrices and Graphs over the ring of integers of some imaginary quadratic fields
• Cyclotomic matrices over real quadratic integer rings
• On the resolution of the sensitivity conjecture
• Integer symmetric matrices of small spectral radius and small Mahler measure
• Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs
• The Cauchy interlace theorem for symmetrizable matrices
• Ex Post Conditions for the Exactness of Optimal Power Flow Conic Relaxations
• A Test Matrix for an Inverse Eigenvalue Problem
• Lehmer's conjecture for Hermitian matrices over the Eisenstein and Gaussian integers
• Energy-conserving intermittent-contact motion in complex models