Woodbury Matrix Identity

Formula is also known ad Woodbury formula.

Sherman-Morrison formula

This formula is a special case of Woodbury formula

Theorem

• $A \in \mathbb{R}^{n \times n}$,

\[\begin{equation} (A + uv^{\mathrm{T}})^{-1} = A^{-1} - \frac{ A^{-1}uv^{\mathrm{T}}A^{-1} }{ 1 + v^{\mathrm{T}}A^{-1}u } \end{equation} .\]

proof

$\Box$

Reference

• Woodbury matrix identity - Wikipedia
• Sherman–Morrison formula - Wikipedia