Interlacing inequalities for totally nonnegative matrices

Suppose lambda (1) greater than or equal to ... greater than or equal to lambda (n) greater than or equal to 0 are the eigenvalues of an n x n totally nonnegative matrix, and lambda (1) greater than or equal to ... greater than or equal to lambda (k) are the eigenvalues of a k x k principal submatrix. A short proof is given of the interlacing inequalities: lambda (i) greater than or equal to lambda (i) greater than or equal to lambda (i+n-k), i = 1,..., k. It is shown that if k = 1, 2, n - 2, n - 1, lambda (i) and lambda (j) are nonnegative numbers satisfying the above inequalities, then there exists a totally nonnegative matrix with eigenvalues lambda (i) and a submatrix with eigenvalues lambda (j). For other values of k, such a result does not hold. Similar results for totally positive and oscillatory matrices are also considered. (C) 2002 Elsevier Science Inc. All rights reserved.