Woodbury Matrix Identity - Direct Proof

Direct Proof

Just check that times the RHS of the Woodbury identity gives the identity matrix:

\begin{align} &\left(A+UCV \right) \left \\ & \quad = I + UCVA^{-1} - (U+UCVA^{-1}U)(C^{-1} + VA^{-1}U)^{-1}VA^{-1} \\ & \quad = I + UCVA^{-1} - UC(C^{-1}+ VA^{-1}U)(C^{-1} + VA^{-1}U)^{-1}VA^{-1} \\ & \quad = I + UCVA^{-1} - UCVA^{-1} = I \end{align}

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