I want to efficiently compute the inverse of
$$T = RVD_1V^T F$$
where $D_1 \in \mathbb R^{k\times k}$ is a diagonal matrix, $V \in \mathbb R^{k\times k}$, $R \in \mathbb R^{n\times k}$, $F \in \mathbb R^{k\times n}$ and $k > n$.
Is it possible to write $T$ or $VD_1V^T$ as the sum of diagonal and low-rank components? I think there is a way to write it, but I wasn't able to find it.
If I can write $T$ as $D + LL^T$ where $D$ is a diagonal matrix, then I can efficiently take the inverse of $D + LL^T$ using approximating diagonal of inverse sum of low rank and diagonal matrices or Inverse of the sum of a invertible matrix with known Cholesky-decomposion and diagonal matrix. Or is there any other method to efficiently compute the inverse of $T$?
