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Sherman-Morrison Formula

The Sherman-Morrison formula is a formula that allows a perturbed matrix to be computed for a change to a given matrix A. If the change can be written in the form

u tensor v
(1)

for two vectors u and v, then the Sherman-Morrison formula is

(A+u tensor v)^(-1)=A^(-1)-((A^(-1)u) tensor (v·A^(-1)))/(1+lambda),
(2)

where

lambda=v·A^(-1)u.
(3)

See also

Woodbury Formula

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References

Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, p. 51, 1996.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.

Referenced on Wolfram|Alpha

Sherman-Morrison Formula

Cite this as:

Weisstein, Eric W. "Sherman-Morrison Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Sherman-MorrisonFormula.html

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