Let $Z \in \Bbb R^{n \times p}$, where $n > p$, and let $X := Z_{-ij} \in \Bbb R^{n \times (p-2)}$ denote the matrix in which the $i$ and $j$-th columns in $Z$ are omitted. Given the eigendecomposition of $Z Z^T = U D U^T$, is there any way to efficiently (meaning I don't want to compute the eigendecomposition of $X^TX,XX^T$) compute $E^{-1}$ (assume $B, C \in \Bbb R^{2\times 2}$ s.t. $E^{-1}$ exists), where
$$ E= C \otimes X^T X + B \otimes I_{p-2} = \begin{bmatrix} c_{11} X^T X + b_{11} I_{p-2} & c_{12} X^T X + b_{12} I_{p-2} \\ c_{21} X^T X + b_{21} I_{p-2} & c_{22} X^T X + b_{22} I_{p-2} \end{bmatrix} $$
I tried inverse block matrix formula to utilize the low rank perturbation but it didn't work. Any hints would be appreciated!