3  Woodbury identity

3.1 Result

If we have matrices \(A_{n \times n}, U_{n\times k}, C_{k \times k}, V_{k \times n}\), then

\[( A + UCV)^{-1} = A^{-1} - A^{-1} U (C^{-1} + VA^{-1}U)^{-1} VA^{-1} \]

3.2 Derivation

Consider matrices \(X\) and \(Y\).

\[\begin{align*} X(I + YX) &= (I+XY)X\\ (I+XY) ^{-1} X &= X(I + YX) ^{-1} \end{align*}\]

\[\begin{align*} I &= (I+P)(I+P)^{-1} \\ I &= (I+P)^{-1} + P(I+P)^{-1} \\ (I+P)^{-1} &= I - P(I+P)^{-1} \quad \text{(Rearranging terms)}\\ (I + YX)^{-1} &= I - YX(I+YX)^{-1} \quad \text{(Substituting } P = YX \text{)}\\ (I + YX)^{-1} &= I - Y (I+XY)^{-1}X \quad \text{(Using (1)) } \end{align*}\]

We substitue \(Y := A^{-1} U\) and \(X := C V\)

\[\begin{align*} (I + A^{-1} UCV)^{-1} &= I - A^{-1} U (I + CVA^{-1}U)^{-1} CV\\ (A^{-1} A + A^{-1} UCV)^{-1} &= I - A^{-1} U (C C^{-1} + CVA^{-1}U)^{-1} CV\\ [A^{-1}( A + UCV)]^{-1} &= I - A^{-1} U [ C (C^{-1} + VA^{-1}U)]^{-1} CV\\ ( A + UCV)^{-1} A &= I - A^{-1} U (C^{-1} + VA^{-1}U)^{-1} C^{-1} CV\\ ( A + UCV)^{-1} &= A^{-1} - A^{-1} U (C^{-1} + VA^{-1}U)^{-1} VA^{-1} \end{align*}\]