Results 1 to 4 of 4

Thread: Inverse of a Spectral Matrix

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    5

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    7
    Awards
    2
    Is $\displaystyle \{\mathbf{v}_{i}\}$ an orthonormal set?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    5
    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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    7
    Awards
    2
    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?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Finding the inverse of a matrix using it's elementary matrix?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Mar 7th 2011, 06:08 PM
  2. [SOLVED] Derivative of a matrix inverse and matrix determinant
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 24th 2011, 08:18 AM
  3. Spectral Decomposition of a Matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 17th 2010, 11:03 AM
  4. Replies: 3
    Last Post: Mar 1st 2010, 06:22 AM
  5. Inverse of a 3 by 3 matrix
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: Sep 14th 2009, 01:41 AM

Search Tags


/mathhelpforum @mathhelpforum