Weak interlacing properties of totally positive matrices (Q1086651)

A real valued \(n\times n\) matrix A is totally positive of order k if all \(j\times j\) minors of A are nonnegative for \(j=1,...,k\). For an \(n\times n\) matrix A let \(\tilde A_ k\) be an (n-1)\(\times (n-1)\) matrix obtained from A by deleting the kth row and column of A and let \(\lambda_ 1(A),\lambda_ 2(A),...,\lambda_ n(A)\) be the eigenvalues of A arranged in order of decreasing magnitude, \(| \lambda_ 1(A)\geq | \lambda_ 2(A)| \geq...\geq | \lambda_ n(A)|\). Then the author shows that \(\lambda_ 1(\tilde A_ k)\geq \lambda_ 2(A)\), \(k=1,...,n\), if A is totally positive of order 2. And \(\lambda_{n- 1}(\tilde A_ k)\geq \lambda_ n(A)\), \(k=1,...,n\), if all (n-2) and (n- 1), and n ordered minors of A are non-negative. These results are partial extensions of the well-known Cauchy interlacing inequalities for symmetric operators to totally positive matrices.