Woodbury matrix identity

For an invertible $n\times n$ matrix $A$, an invertible $k\times k$ matrix $C$, and compatible $U,V$, the Woodbury matrix identity isprovided $C^{-1}+V{A}^{-1}U$ 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.