= Woodbury matrix identity
{c}
= Sherman–Morrison–Woodbury
{c}
{synonym}
For an invertible $n\times n$ <matrix> $A$, an invertible $k\times k$ <matrix> $C$, and compatible $U,V$, the <Woodbury matrix identity> is
$$
(A+UCV)^{-1}=A^{-1}-A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1},
$$
provided $C^{-1}+VA^{-1}U$ is invertible. Multiplying the right side by $A+UCV$ verifies the formula directly.
A rank-$k$ update therefore requires inversion of a $k\times k$ correction rather than a new $n\times n$ inverse. For $k=1$, it reduces to the <Sherman–Morrison formula>. In implementations, occurrences of $A^{-1}U$ are usually obtained by solving <linear systems>. The smaller inverse must still exist; the identity does not make every low-rank update invertible.
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