Schur complement formula for a diagonal resolvent entry ($G_{ii}=(X_{ii}-z-x_i^{\ast }{G}^{(i)}{x}_i{)}^{-1}$)

For a Hermitian matrix, let $x_i$ be its $i$th column without the diagonal entry and let $G^{(i)}=(X^{(i)}-zI{)}^{-1}$. Solve the two block equations of the inverse against the $i$th coordinate vector to obtain this formula. Both full and minor inverses must exist; nonreal $z$ guarantees that. In the real symmetric case $x_i^{\ast }$ may be replaced by $x_i^T$.