I'm working on a problem that involves multiplying the inverse of a spectral matrix by a matrix containing a subset of that matrices spectral components. Specifically:

$\displaystyle

{\bf S}_N = \sum\limits_{i = 1}^N {\lambda _i {\bf v}_i {\bf v}_i ^H }

$

$\displaystyle

{\bf S}_{N - 1} = \sum\limits_{i = 1}^{N - 1} {\lambda _i {\bf v}_i {\bf v}_i ^H }

$

How can I determine

$\displaystyle

\left[ {{\bf S}_N } \right]^{ - 1} S_{N - 1} = ?

$

S is a covariance matrix.

$\displaystyle {\lambda _i }$ is the ith eigenvalue.

$\displaystyle {{\bf v}_i } $ is the ith eigenvector.

Also, can the inverse of a specral matrix be writen in terms of the inverse of each spectral component $\displaystyle {\lambda _i {\bf v}_i {\bf v}_i ^H }$ somehow?

Thank you