Woodbury formula

This is a widely used formula for inverses. It goes by several names. The longest one is Sherman-Woodbury-Morrison formula, but it goes by any subset of these three names in any order. In its most general form, it can be expressed as

$$(A+BC)^{-1} = A^{-1}-A^{-1} B (I+C A^{-1}B)^{-1}CA^{-1}$$.

A special case for vectors is:

$$(I+cd^T)^{-1}=I-\frac{cd^T}{1+d^Tc}$$

Another special case is

$$(A+cd^T)^{-1}=A^{-1}-\frac{A^{-1}cd^TA^{-1}}{1+d^TA^{-1}c}$$

This is often called the Sherman-Morrison rank-one update formula.