Suppose we have matrices $P\in \mathbb{R}^{n\times n}$ and $G_i\in \mathbb{R}^{m\times m}$ for $i=1,\ldots,n$ which are all symmetric positive semi-definit. We have that $m\ll n$.
We define the symmetric positive semi-definit matrix $Q\in \mathbb{R}^{n\times n}$ with entries $$Q_{ij}=trace(G_i G_j).$$
I'm interested in computing $$[P^{-1} + Q]^{-1}$$ without explicitly invert the $n$ times $n$ matrix $P^{-1} + Q$. However, since $Q$ has full rank, I don't have much hope to do this (matrix inversion lemma).
Does anybody have an approach how I can avoid to explicitly compute this inversion?
Thanks.