Sherman–Morrison formula

The Sherman - Morrison - Woodbury formula (after Jack Sherman, Winifred J. Morrison and Max A. Woodbury ) linear algebra is an explicit representation of the inverse of a regular matrix after a change from a low-ranking. This is interesting, for example, for quasi -Newton method and the change of basis in the simplex method.

In numerical methods can lead to stability problems using the formula, which is why alternatives are preferable.

Change of rank 1

With two vectors, the product is a matrix and has rank 1

This can be checked by elementary.

The formula carries over directly to rank-1 changes an arbitrary regular matrix:

This implies that the matrix is ​​invertible when the denominator in the above formula does not disappear.

Change of rank

For two matrices, the formula generalizes in the following way: