Previously I wrote about the LDU decomposition and the Schur complement. These can be further used to derive the Shermanâ€“Morrisonâ€“Woodbury formula, otherwise known as the matrix inversion lemma, for inverting a matrix. As shown in the previous post, a UDL and LDU are two ways of factorizing a matrix:

Now consider taking the inverse of the matrices above, yielding

Multiplying the matrices on the RHS yields

These two results can be used to form neat expressions for the inverse of a partitioned block matrix. It follows that

The equality holds for each block or element, so two expressions can be found for the Woodbury matrix inverse formula, namely:

and

where the dot notation corresponds to the Schur complement i.e.

An application of the Woodbury matrix inverse can be found in deriving conditional distributions for multivariate normals.

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