# Thread: Inverse of a Spectral Matrix

1. ## Inverse of a Spectral Matrix

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

2. Is $\displaystyle \{\mathbf{v}_{i}\}$ an orthonormal set?

3. Yes $\displaystyle \left\{ {{\bf v}_{\bf i} } \right\}$ is an orthonormal set.

Also, in genral I'm interested in figuring out how to solve

$\displaystyle \left[ {S_N } \right]^{ - 1} S_{N - k} = ?$

for an arbitrary 0 < k < N

Thanks

4. Well then, it sounds like you've got yourself a diagonalized matrix. You can write

$\displaystyle S_{N}=P^{-1}DP,$ where

$\displaystyle P=[\mathbf{v}_{1}\;\mathbf{v}_{2}\;\dots\;\mathbf{v}_ {N}],$ and

$\displaystyle \displaystyle D_{ij}=\delta_{ij}\lambda_{i}.$

In that case, you have $\displaystyle S_{N}^{-1}=(P^{-1}DP)^{-1}=P^{-1}D^{-1}P.$

Now $\displaystyle D^{-1}$ is easy to calculate, since

$\displaystyle (\delta_{ij}\dfrac{1}{\lambda_{i}})(\delta_{ij}\la mbda_{i})=\delta_{ij}^{2}=\delta_{ij}.$

Therefore,

$\displaystyle D_{ij}^{-1}=\delta_{ij}\dfrac{1}{\lambda_{i}}.$

Make sense?