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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

{\bf u}\otimes{\bf v}

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

({\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},


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

See also Woodbury Formula


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.

© 1996-9 Eric W. Weisstein