Sherman-Morrison Formula

A formula which allows the new Matrix to be computed for a small change to a Matrix ${\hbox{\sf A}}$ . If the change can be written in the form

$\begin{displaymath} {\bf u}\otimes{\bf v} \end{displaymath}$

for two vectors ${\bf u}$ and ${\bf v}$ , then the Sherman-Morrison formula is

$\begin{displaymath} ({\hbox{\sf A}}+{\bf u}\otimes{\bf v})^{-1}={\hbox{\sf A}}^{... ...f u})\otimes({\bf v}\cdot{\hbox{\sf A}}^{-1})\over 1+\lambda}, \end{displaymath}$

where

$\begin{displaymath} \lambda\equiv {\bf v}\cdot{\hbox{\sf A}}^{-1}{\bf u}. \end{displaymath}$

See also Woodbury Formula

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Sherman-Morrison Formula.'' In Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 65-67, 1992.