An observation about submatrices. (English) Zbl 1189.60041

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.


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