So, lets assume that we have two matrices given to us:
$A \in \mathbb{\Re}^{M \times K}$: just some dense, real, arbitrary valued matrix.
$L \in \Re^{K \times K}$: A positive diagonal matrix
Now, lets say we have the following situation:
$\left( A^T A + L \right)^{-1}$
Is it possible to rephrase this statement in such a way as to make use of the knowledge of $\left( A^T A \right)^{-1}$ so as to avoid this addition inside the inverse?
I'm sorry if this is a little vague :P Just running into an analytical hiccup and trying to figure out how to break this statement apart.
Edit: Would it be possible to analyze this expression using an eigen decomposition? That is, if we say
$A^T A = Q \Lambda Q^{-1}$
Then, if $L$ is diagonal:
$\left( A^T A + L \right) = Q \left( \Lambda + L \right) Q^{-1}$
Which would make
$\left( A^T A + L \right)^{-1} = Q \left( \Lambda + L \right)^{-1} Q^{-1}$
Which might be usable somehow because $\left( \Lambda + L \right)$ is a diagonal matrix and allowing you to calculate the inverse directly by taking $1$ over the diagonal entries, right? Or is this completely off?
Edit 2: @Rcollyer So, I changed around my formulation entirely by only looking at rank 1 updates of $A^T A$ and was able to use the Sherman-Morrison (a form of the Woodbury) formula to break it apart and get at what I wanted :P