An observation about submatrices. (English) Zbl 1189.60041

Chatterjee, Sourav; Ledoux, Michel

Electron. Commun. Probab. 14, 495-500 (2009).

Summary: Let \(M\) be an arbitrary Hermitian matrix of order \(n\), and \(k\) be a positive integer less than \(n\). 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 \(k\times n\) submatrices of \(M\) have almost the same distribution of singular values.

MSC:

60E15	Inequalities; stochastic orderings
15B52	Random matrices (algebraic aspects)
60B20	Random matrices (probabilistic aspects)

Keywords:

random matrix; concentration of measure; empirical distribution; eigenvalue