In my work on Gaussian processes, I encountered the following operation on matrices. Let $K$ be a positive definite $d \times d$ matrix. Define $$ \tilde{K}_{ij} = K_{ij} - \frac{(K_{ki} + K_{li}) (K_{kj} + K_{lj})}{4(K_{kk} + 2K_{kl}+K_{ll})}, \quad i, \, j \in \{ 1, \ldots, d \} \setminus \{ k, l \}. $$ I wonder if this operation is known in the literature and has a name. It seems related to the Schur complement, but I couldn't quite figure how to put it in this context.
Some context. This matrix appears in the calculation of $$ \int_{\mathbb{R}^d} \delta (x_k - x_l) \, e^{-\mathbf{x}^\top K \, \mathbf{x}/2} \, d\mathbf{x}. $$