Let \(A=\left[ \begin{matrix} H\\ B\end{matrix} \begin{matrix} B^*\\ U\end{matrix} \right],H:m\times m\), be an \(n\times n\) Hermitian matrix. If \(\alpha_ 1\leq \alpha_ 2\leq...\leq \alpha_ n\) and \(\theta_ 1\leq \theta_ 2\leq...\leq \theta_ m\) are eigenvalues of A and H respectively, then according to the Cauchy interlacing theorem \(\alpha_ i\leq \theta_ i\leq \alpha_{n-m+i},\quad i=1,2,...,m.\) This paper elaborates on the case when some of the inequalities above become equalities or near- equalities. In particular, it is shown that if \(\alpha_{k+1}<\alpha_ k\) for some k, then \(\sum^{k}_{i=1}\sin^ 2\Phi_ i\leq 1/(\alpha_{k+1}-\alpha_ k)\sum^{k}_{i=1}(\theta_ i-\alpha_ i)\) where \(\Phi_ i\) are the angles between the subspaces spanned by the first k eigenvectors of A and H.