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Hello,The pseudoinverse formula for a matrix A is given by:P =

  1. Oct 29, 2011 #1

    The pseudoinverse formula for a matrix A is given by:

    P = (A[itex]^{T}[/itex]A)[itex]^{-1}[/itex]A[itex]^{T}[/itex]

    I remember knowing this some time ago and this has me worried now....why is (A[itex]^{T}[/itex]A)[itex]^{-1}[/itex] guaranteed to exist? I know it is a square matrix but it could still be degenerate, right?

    Would appreciate any help.

  2. jcsd
  3. Oct 29, 2011 #2


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    Science Advisor

    Re: pseudoinverse

    it's not guaranteed to exist.

    for example, consider A =

    [1 1]
    [0 0].

    the (left) pseudoinverse is useful when you have an nxk matrix (n > k) of rank k. then ATA is an invertible kxk matrix, so

    (ATA)-1AT acts as a left-inverse (to the left-identity) for A, as you can easily verify by computation.

    (in this case, In is merely a left-identity for the nxk matrices, the right-identity is Ik, and these matrices are of different sizes).

    in other words, the pseudo-inverse "acts" like an inverse on a class of matrices which don't have inverses.
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