Schur complement formula for a diagonal resolvent entry
= Schur complement formula for a diagonal resolvent entry
{c}
{title2=$G_{ii}=(X_{ii}-z-x_i^*G^{(i)}x_i)^{-1}$}
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 $x_i^*$ may be replaced by $x_i^T$.