Derivation Via Blockwise Elimination
Deriving the Woodbury matrix identity is easily done by solving the following block matrix inversion problem
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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