On the spread of a hermitian matrix and a conjecture of thompson

We prove some inequalities involving the eigenvalues of an nxn Hermitian matrix and the eigenvalues of the (n−1)x(n−1) principal submatrices. We apply this inequality to generalize a known result on the numerical range to the lth numerical range. The method used yields an elegant proof of the converse to the interlacing theorem, which we include. A counterexample to the quardratic spread inequality conjectured by R. C. Thompson is also given.