Sherman–Morrison formula

In mathematics, in particular linear algebra, the Sherman–Morrison formula,^[1]^[2]^[3] named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix $A$ and the outer product, $uv^{T}$ , of vectors $u$ and $v$ . The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications.^[4]

Statement

Suppose $A$ is an invertible square matrix and $u$ , $v$ are column vectors. Suppose furthermore that $1+v^{T}A^{-1}u\neq 0$ . Then the Sherman–Morrison formula states that

(A+uv^{T})^{-1}=A^{-1}-{A^{-1}uv^{T}A^{-1} \over 1+v^{T}A^{-1}u}.

Here, $uv^{T}$ is the outer product of two vectors $u$ and $v$ . The general form shown here is the one published by Bartlett.^[5]

Application

If the inverse of $A$ is already known, the formula provides a numerically cheap way to compute the inverse of $A$ corrected by the matrix $uv^{T}$ (depending on the point of view, the correction may be seen as a perturbation or as a rank-1 update). The computation is relatively cheap because the inverse of $A+uv^{T}$ does not have to be computed from scratch (which in general is expensive), but can be computed by correcting (or perturbing) $A^{-1}$ .

Using unit columns (columns from the identity matrix) for $u$ or $v$ , individual columns or rows of $A$ may be manipulated and a correspondingly updated inverse computed relatively cheaply in this way.^[6] In the general case, where $A^{-1}$ is a $n$ -by- $n$ matrix and $u$ and $v$ are arbitrary vectors of dimension $n$ , the whole matrix is updated^[5] and the computation takes $3n^{2}$ scalar multiplications.^[7] If $u$ is a unit column, the computation takes only $2n^{2}$ scalar multiplications. The same goes if $v$ is a unit column. If both $u$ and $v$ are unit columns, the computation takes only $n^{2}$ scalar multiplications.

Verification

We verify the properties of the inverse. A matrix $Y$ (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix $X$ (in this case $A+uv^{T}$ ) if and only if $XY=YX=I$ .

We first verify that the right hand side ( $Y$ ) satisfies $XY=I$ .

XY=(A+uv^{T})\left(A^{-1}-{A^{-1}uv^{T}A^{-1} \over 1+v^{T}A^{-1}u}\right)

=AA^{-1}+uv^{T}A^{-1}-{AA^{-1}uv^{T}A^{-1}+uv^{T}A^{-1}uv^{T}A^{-1} \over 1+v^{T}A^{-1}u}

=I+uv^{T}A^{-1}-{uv^{T}A^{-1}+uv^{T}A^{-1}uv^{T}A^{-1} \over 1+v^{T}A^{-1}u}

=I+uv^{T}A^{-1}-{u(1+v^{T}A^{-1}u)v^{T}A^{-1} \over 1+v^{T}A^{-1}u}

Note that $v^{T}A^{-1}u$ is a scalar, so $(1+v^{T}A^{-1}u)$ can be factored out, leading to:

XY=I+uv^{T}A^{-1}-uv^{T}A^{-1}=I.\,

In the same way, it is verified that

YX=\left(A^{-1}-{A^{-1}uv^{T}A^{-1} \over 1+v^{T}A^{-1}u}\right)(A+uv^{T})=I.

Following is an alternate verification of the Sherman–Morrison formula using the easily verifiable identity

(I+wv^{T})^{-1}=I-{\frac {wv^{T}}{1+v^{T}w}}

Let $u=Aw$ and $A+uv^{T}=A\left(I+wv^{T}\right)$ , then

(A+uv^{T})^{-1}=(I+wv^{T})^{-1}{A^{-1}}=\left(I-{\frac {wv^{T}}{1+v^{T}w}}\right)A^{-1}

Substituting $w={{A}^{{-1}}}u$ gives

(A+uv^{T})^{-1}=\left(I-{\frac {A^{-1}uv^{T}}{1+v^{T}A^{-1}u}}\right)A^{-1}={A^{-1}}-{\frac {A^{-1}uv^{T}A^{-1}}{1+v^{T}A^{-1}u}}

Generalization

Assume that we have some (n × n) matrix, A. U is a (n × k) matrix and V is a (k × n) matrix, B = A + UV. Then,

{B^{-1}}={A^{-1}}-{A^{-1}}U{\left({{I_{k}}+V{A^{-1}}U}\right)^{-1}}V{A^{-1}}

References

↑ Sherman, Jack; Morrison, Winifred J. (1949). "Adjustment of an Inverse Matrix Corresponding to Changes in the Elements of a Given Column or a Given Row of the Original Matrix (abstract)". Annals of Mathematical Statistics. 20: 621. doi:10.1214/aoms/1177729959.
↑ Sherman, Jack; Morrison, Winifred J. (1950). "Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix". Annals of Mathematical Statistics. 21 (1): 124–127. doi:10.1214/aoms/1177729893. MR 35118. Zbl 0037.00901.
↑ Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), "Section 2.7.1 Sherman–Morrison Formula", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
↑ Hager, William W. (1989). "Updating the inverse of a matrix". SIAM Review. 31 (2): 221–239. doi:10.1137/1031049. JSTOR 2030425. MR 997457.
1 2 Bartlett, Maurice S. (1951). "An Inverse Matrix Adjustment Arising in Discriminant Analysis". Annals of Mathematical Statistics. 22 (1): 107–111. doi:10.1214/aoms/1177729698. MR 40068. Zbl 0042.38203.
↑ Langville, Amy N.; and Meyer, Carl D.; "Google's PageRank and Beyond: The Science of Search Engine Rankings", Princeton University Press, 2006, p. 156
↑ Update of the inverse matrix by the Sherman–Morrison formula

External links

Weisstein, Eric W. "Sherman–Morrison formula". MathWorld.

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