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Reconstruct A from its reduced eigenparis

  1. Jul 15, 2014 #1
    Hi all,

    Suppose I have a matrix [itex] A_{N\times N}[/itex]. I compute its eigenmodes

    [itex] A V = V \Lambda[/itex].

    [itex] V, \Lambda[/itex] are eigenvectors and eigenvalues of size [itex] N\times N [/itex]. The eigenvalues are descending.

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

    [itex] A V_{N \times n} = V_{N \times n} \Lambda_{n \times n}[/itex].

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

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

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

    it will not work.

    So my question is: how can I reliably reconstruct the original matrix by using its reduced eigenmodes? Thanks!!
     
    Last edited: Jul 15, 2014
  2. jcsd
  3. Jul 15, 2014 #2

    AlephZero

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    I don't think you can do this for an arbitrary matrix ##A##. It's not clear what you mean by ##V_{N\times m}^{-1}## for a non-square matrix ##V_{N\times m}##.

    You can do something similar with the singular value decomposition ##A = U\Sigma V^T## where ##U## and ##V## are unitary matrices.

    You can also do what you want if ##A## is Hermitian, because ##V^{-1} = V^T##.
     
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