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Math Help - Inverse of a Spectral Matrix

  1. #1
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    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:

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

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

    How can I determine
     <br />
\left[ {{\bf S}_N } \right]^{ - 1} S_{N - 1} = ?<br />

    S is a covariance matrix.
    {\lambda _i } is the ith eigenvalue.
     {{\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 {\lambda _i {\bf v}_i {\bf v}_i ^H } somehow?

    Thank you
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  2. #2
    A Plied Mathematician
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    Is \{\mathbf{v}_{i}\} an orthonormal set?
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  3. #3
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    Yes \left\{ {{\bf v}_{\bf i} } \right\} is an orthonormal set.

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

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

    for an arbitrary 0 < k < N

    Thanks
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  4. #4
    A Plied Mathematician
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    Well then, it sounds like you've got yourself a diagonalized matrix. You can write

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

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

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

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

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

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

    Therefore,

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

    Make sense?
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