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Math Help - Reconstruct a matrix from its reduced eigenparis

  1. #1
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    Reconstruct a matrix from its reduced eigenparis

    Hi all,


    Suppose I have a matrix  A_{N\times N}. I compute its eigenmodes




     A V = V \Lambda.




     V, \Lambda are eigenvectors and eigenvalues of size  N\times N . The eigenvalues are put in a descending order.




    Now I cut off several eigenmodes (the ones having small value), it becomes




     A V_{N \times n} = V_{N \times n} \Lambda_{n \times n}.




    What I want is to reconstruct A from  V_{N \times n}, \Lambda_{n \times n}.




    It seems that in Matlab, if I just modify the above equation




     A  = V_{N \times n} \Lambda_{n \times n}V^{-1}_{N \times n},




    it will not work.




    So my question is: how can I reliably reconstruct the original matrix by using its reduced eigenmodes? Thanks!!
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  2. #2
    MHF Contributor
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    Re: Reconstruct a matrix from its reduced eigenparis

    I'm not sure what you mean by "reliably". Your reconstruction is going to be lossy.

    That being said if you've decomposed your original matrix as

    $A=Q \Lambda Q^{-1}$

    where $Q$ is the matrix of eigenvector columns and

    $\Lambda$ is the diagonal matrix of eigenvalues

    you should be able to just zero out the eigenvalues below some fidelity threshold, thus forming $\Lambda_{reduced}$.

    Then

    $A_{reduced}=Q \Lambda_{reduced} Q^{-1}$
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