Interlacing inequalities for totally nonnegative matrices Original Research Article

Suppose λ1greater-or-equal, slantedcdots, three dots, centeredgreater-or-equal, slantedλngreater-or-equal, slanted0 are the eigenvalues of an n×n totally nonnegative matrix, and image are the eigenvalues of a k×k principal submatrix. A short proof is given of the interlacing inequalities: image It is shown that if k=1,2,n−2,n−1, λi and image are nonnegative numbers satisfying the above inequalities, then there exists a totally nonnegative matrix with eigenvalues λi and a submatrix with eigenvalues image . For other values of k, such a result does not hold. Similar results for totally positive