Schur complement formula for a diagonal entry of $(GG^\top)^{-1}$: $((GG^\top)^{-1})_{ii} = 1/\|(I-P_i)g_i\|^2$

Let $G \in \mathbb{R}^{(n+1)\times k}$ be a real matrix, fix a row index $i \in \{0,\dots,n\}$, let $g = G_{i,:} \in \mathbb{R}^k$ be the $i$-th row of $G$, and let $H \in \mathbb{R}^{n\times k}$ be the matrix obtained from $G$ by deleting row $i$ (its rows are $G_{i',:}$, $i' \neq i$, in their original order). Assume that the Gram matrix $HH^\top \in \mathbb{R}^{n\times n}$ is invertible. Then

where $P_H = H^\top (HH^\top)^{-1} H$ is the orthogonal projector of $\mathbb{R}^k$ onto the row space of $H$, so that $(I-P_H)g$ is the residual of $g$ after projection onto the span of the other rows. The quantity $g^\top g - (Hg)^\top (HH^\top)^{-1}(Hg)$ is the Schur complement of the block $HH^\top$ in $GG^\top$.

This is the deterministic linear-algebra step in the classical computation of the mean of an inverse Wishart matrix: it expresses each diagonal entry of $(GG^\top)^{-1}$ through the squared distance from one row to the span of the others, which for a Gaussian matrix has a chi-square law.

Formalization Note. Lean's matrix inverse is total ($A^{-1} = 0$ when $\det A = 0$) and $0^{-1} = 0$ in $\mathbb{R}$; the identity holds under the sole hypothesis $\det(HH^\top) \ne 0$, including the degenerate case where $g$ lies in the row space of $H$ (then both sides are $0$). The matrix $H$ is written G.submatrix i.succAbove id.