An observation about submatrices

TL;DR

The paper demonstrates that for large k, most principal submatrices of a Hermitian matrix have similar eigenvalue distributions, and most k x n submatrices share similar singular value distributions, using probabilistic and combinatorial methods.

Contribution

It establishes that eigenvalue and singular value distributions are nearly uniform across most submatrices of a Hermitian matrix when the submatrix size is large.

Findings

Eigenvalue distributions are nearly identical for most principal submatrices of large size.

Singular value distributions are similar across most k x n submatrices.

Uses random walks on symmetric groups and measure concentration techniques.

Abstract

Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of M of order k. The proof uses results about random walks on symmetric groups and concentration of measure. In a similar way, we also show that almost all k x n submatrices of M have almost the same distribution of singular values.