So suppose I have the following nonsingular matrix $G$: $$ G = Z + K^T K $$ Where $Z$ is of dimensions $m \times m$ and $K$ is of dimensions $n \times m$ leaving us with $G$ having dimensions $m \times m$. Suppose also that the inverse of $Z$ is readily available to us. Does anyone know of an inverse identity or some other shortcut formula that would help me compute the inverse of $G$ faster? Directly solving for $G^{-1}$ would be of order $O(m^3)$ and I'm expecting $m$ to be really high. I tried Woodbury's identity formula but that just left me with the inversion of an $n \times n$ matrix which is actually worse (because in most applications I expect $n$ to be much higher than $m$).
EDIT: I found the following formula in Methods of Multivariate Analysis by Rencher & Christensen that seems promising: $$ (B + cc^T)^{-1} = B^{-1} - \frac{B^{-1} cc^T B^{-1}}{1 + c^T B^{-1} c} $$ However, this seems to only apply for a vector $c$, and in my case I have a matrix $K$. Not sure if this helps anyone but I thought I'd include it in case someone has seen a similar application.