Interlacing Inequalities for Totally Nonnegative Matrices

Suppose λ1 ≥ · · · ≥ λn ≥ 0 are the eigenvalues of an n × n totally nonnegative matrix, and λ̃1 ≥ · · · ≥ λ̃k are the eigenvalues of a k × k principal submatrix. A short proof is given of the interlacing inequalities: λi ≥ λ̃i ≥ λi+n−k, i = 1, . . . , k. It is shown that if k = 1, 2, n− 2, n− 1, λi and λ̃j are nonnegative numbers satisfying the above inequalities, then there exists a totally nonnegative matrix with eigenvalues λi and a submatrix with eigenvalues λ̃j . For other values of k, such a result does not hold. Similar results for totally positive and oscillatory matrices are also considered.