## 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.

### MSC:

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