Interlacing inequalities for totally nonnegative matrices

AbstractSuppose 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