# Inverse Matrices on Subspaces

1. Sep 28, 2009

### Kreizhn

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 $\mathbb C^9$, 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 $v_i, i = 0, \ldots 3$ then the $9 \times 9$ matrix is as follows:
$$M = \sum_{i=0}^3 v_i^T v_i$$
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 $\{ v_i \}$. 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.