Abstract

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

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