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 $A\in \mathbb {R} ^{n\times n}$ is an invertible square matrix and $u$ , $v\in \mathbb {R} ^{n}$ are column vectors. Then $A+uv^{T}$ is invertible iff $1+v^{T}A^{-1}u\neq 0$ . If $A+uv^{T}$ is invertible, then its inverse is given by

(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]

Proof

( $\Leftarrow$ ) To prove that the backward direction ( $1+v^{T}A^{-1}u\neq 0\Rightarrow A+uv^{T}$ is invertible with inverse given as above) is true, 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 $XY=I$ .

{\begin{aligned}XY&=(A+uv^{T})\left(A^{-1}-{A^{-1}uv^{T}A^{-1} \over 1+v^{T}A^{-1}u}\right)\\[6pt]&=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}\\[6pt]&=I+uv^{T}A^{-1}-{uv^{T}A^{-1}+uv^{T}A^{-1}uv^{T}A^{-1} \over 1+v^{T}A^{-1}u}\\[6pt]&=I+uv^{T}A^{-1}-{u(1+v^{T}A^{-1}u)v^{T}A^{-1} \over 1+v^{T}A^{-1}u}\\[6pt]&=I+uv^{T}A^{-1}-uv^{T}A^{-1}&&1+v^{T}A^{-1}u{\text{ is a scalar}}\\[6pt]&=I\end{aligned}}

In the same way, one can verify that:

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

( $\Rightarrow$ ) To prove the reverse direction, we suppose that $u\neq 0$ , otherwise the result is trivial. Then,

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

Since $A+uv^{T}$ is assumed invertible, $(A+uv^{T})A^{-1}$ is invertible as the product of invertible matrices. Thus, by our assumption that $u\neq 0$ , we have that $(A+uv^{T})A^{-1}u\neq 0$ . By the identity above, this means that $(1+v^{T}A^{-1}u)u\neq 0$ , and hence $1+v^{T}A^{-1}u\neq 0$ , as was to be shown.

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.

This formula also has application in theoretical physics. Namely, in quantum field theory, one uses this formula to calculate the propagator of a spin-1 field ^[8]. The inverse propagator (as it appears in the Lagrangian) has the form $(A+uv^{T})$ . One uses the Sherman-Morrison formula to calculate the inverse (satisfying certain time-ordering boundary conditions) of the inverse propagator - or simply the (Feynman) propagator - which is needed to perform any perturbative calculation^[9] involving the spin-1 field.

Alternative verification

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,\quad {\text{and}}\quad 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 (Woodbury matrix identity)

Given a square invertible $n\times n$ matrix $A$ , an $n \times k$ matrix $U$ , and a $k\times n$ matrix $V$ , let $B$ be an $n\times n$ matrix such that $B=A+UV$ . Then, assuming $\left(I_{k}+VA^{-1}U\right)$ is invertible, we have

B^{-1}=A^{-1}-A^{-1}U\left(I_{k}+VA^{-1}U\right)^{-1}VA^{-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&ndash, 127. doi:10.1214/aoms/1177729893. MR 0035118. 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&ndash, 239. doi:10.1137/1031049. JSTOR 2030425. MR 0997457.
1 2 Bartlett, Maurice S. (1951). "An Inverse Matrix Adjustment Arising in Discriminant Analysis". Annals of Mathematical Statistics. 22 (1): 107&ndash, 111. doi:10.1214/aoms/1177729698. MR 0040068. 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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[SM1949-1] 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.

[SM1950-2] 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&ndash, 127. doi:10.1214/aoms/1177729893. MR 0035118. Zbl 0037.00901.

[PFTV1992-3] 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-4] Hager, William W. (1989). "Updating the inverse of a matrix". SIAM Review. 31 (2): 221&ndash, 239. doi:10.1137/1031049. JSTOR 2030425. MR 0997457.

[B1951-5] 1 2 Bartlett, Maurice S. (1951). "An Inverse Matrix Adjustment Arising in Discriminant Analysis". Annals of Mathematical Statistics. 22 (1): 107&ndash, 111. doi:10.1214/aoms/1177729698. MR 0040068. Zbl 0042.38203.

[6] Langville, Amy N.; and Meyer, Carl D.; "Google's PageRank and Beyond: The Science of Search Engine Rankings", Princeton University Press, 2006, p. 156

[7] Update of the inverse matrix by the Sherman–Morrison formula

[8] ↑

[9] ↑