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Inverse Matrices on Subspaces

  1. Sep 28, 2009 #1
    1. The problem statement, all variables and given/known data
    This is for a much larger question I'm working on.

    I have four linearly independent vectors in [itex] \mathbb C^9 [/itex], and hence they span a four dimensional space. Now I have a matrix that is composed out of their outerproduct, namely, if we let the vectors be [itex] v_i, i = 0, \ldots 3 [/itex] then the [itex] 9 \times 9 [/itex] matrix is as follows:
    [tex] M = \sum_{i=0}^3 v_i^T v_i [/tex]
    where the T is transpose.
    Now in general, an inverse to this matrix does not exist. However, I have been informed that M has an inverse on the four-dimensional subspace spanned by [itex] \{ v_i \} [/itex]. What I want to know is how do I find the form of this matrix?

    3. The attempt at a solution
    I can easily find the four dimesional subspace, the issue is somehow "reducing" M into the space, finding the inverse (which is easy), and then making it 9x9 again. I honestly don't know how to do this at all.
  2. jcsd
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