  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.
