An Observation about Submatrices

Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer ≤ 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 × n submatrices of M have almost the same distribution of singular values. Let M be a square matrix of order n. For any two sets of integers i1, . . . , ik and j1, . . . , jl between 1 and n, M(i1, . . . , ik; j1, . . . , jl) denotes the submatrix of M formed by deleting all rows except rows i1, . . . , ik, and all columns except columns j1, . . . , jl. A submatrix like M(i1, . . . , ik; i1, . . . , ik) is called a principal submatrix. For a Hermitian matrixM of order n with eigenvalues λ1, . . . , λn (repeated by multiplicities), let FM denote the empirical spectral distribution function of M , that is, FM (x) := #{i : λi ≤ x} n . The following result shows that given 1 k ≤ n and any Hermitian matrix M of order n, the empirical spectral distribution is almost the same for almost every principal submatrix of M of order k. Theorem 1. Take any 1 ≤ k ≤ n and a Hermitian matrix M of order n. Let A be a principal submatrix of M chosen uniformly at random from the set of all k × k principal submatrices of M . Let F be the expected spectral distribution function of A, that is, F (x) = EFA(x). Then for each r ≥ 0, P(‖FA − F‖∞ ≥ k−1/2 + r) ≤ 12 √ ke−r √ . Consequently, we have E‖FA − F‖∞ ≤ 13 + √ 8 log k √ k . 2000 Mathematics Subject Classification. 60E15, 15A52.