Woodbury Matrix Identity - Derivation Via Blockwise Elimination

Derivation Via Blockwise Elimination

Deriving the Woodbury matrix identity is easily done by solving the following block matrix inversion problem

\begin{bmatrix} A & U \\ V & -C^{-1} \end{bmatrix}\begin{bmatrix} X \\ Y \end{bmatrix} = \begin{bmatrix} I \\ 0 \end{bmatrix}.

Expanding, we can see that the above reduces to and, which is equivalent to . Eliminating the first equation, we find that, which can be substituted into the second to find . Expanding and rearranging, we have, or . Finally, we substitute into our, and we have . Thus,

We have derived the Woodbury matrix identity.

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