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| In [[mathematics]], in particular [[linear algebra]], the '''Sherman–Morrison formula''',<ref name="SM1949">{{cite journal
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| |first=Jack |last=Sherman
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| |first2=Winifred J. |last2=Morrison
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| |title=Adjustment of an Inverse Matrix Corresponding to Changes in the Elements of a Given Column or a Given Row of the Original Matrix (abstract)
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| |journal=Annals of Mathematical Statistics
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| |volume=20 |pages=621 |year=1949
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| |doi=10.1214/aoms/1177729959 }}</ref><ref name="SM1950">{{cite journal
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| |first=Jack |last=Sherman
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| |first2=Winifred J. |last2=Morrison
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| |title=Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix
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| |journal=[[Annals of Mathematical Statistics]]
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| |volume=21 |issue=1 |pages=124–127 |year=1950
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| |doi=10.1214/aoms/1177729893 |mr=35118 | zbl=0037.00901 }}</ref><ref name="PFTV1992">{{Citation
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| |last1=Press |first1=William H.
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| |last2=Teukolsky |first2=Saul A.
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| |last3=Vetterling |first3=William T.
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| |last4=Flannery |first4=Brian P.
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| |title=Numerical Recipes: The Art of Scientific Computing |edition=3rd |year=2007 |publisher=Cambridge University Press |publication-place=New York
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| |isbn=978-0-521-88068-8
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| |chapter=Section 2.7.1 Sherman–Morrison Formula
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| |chapter-url=http://apps.nrbook.com/empanel/index.html?pg=76 }}</ref> named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an [[inverse matrix|invertible]] [[matrix (mathematics)|matrix]] <math>A</math>
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| and the [[outer product]], <math>u v^T</math>, of [[vector (mathematics)|vectors]] <math>u</math> and <math>v</math>. The Sherman–Morrison formula is a special case of the [[Woodbury matrix identity|Woodbury formula]].
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| Though named after Sherman and Morrison, it appeared already in earlier publications.<ref name="hager">{{cite journal
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| |first=William W. |last=Hager
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| |title=Updating the inverse of a matrix
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| |journal=SIAM Review
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| |volume=31 |year=1989 |pages=221–239 |issue=2
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| |doi=10.1137/1031049 |mr=997457 |jstor=2030425 }}</ref>
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| == Statement ==
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| Suppose <math>A</math> is an [[invertible]] [[square matrix]] and <math>u</math>, <math>v</math> are [[Vector (geometric)|vectors]]. Suppose furthermore that <math>1 + v^T A^{-1}u \neq 0</math>. Then the Sherman–Morrison formula states that
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| :<math>(A+uv^T)^{-1} = A^{-1} - {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}.</math>
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| Here, <math>uv^T</math> is the [[outer product]] of two vectors <math>u</math> and <math>v</math>. The general form shown here is the one published by Bartlett.<ref name="B1951">{{cite journal
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| |first=Maurice S. |last=Bartlett
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| |title=An Inverse Matrix Adjustment Arising in Discriminant Analysis
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| |journal=[[Annals of Mathematical Statistics]]
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| |volume=22 |issue=1 |pages=107–111 |year=1951
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| |doi=10.1214/aoms/1177729698 |mr=40068 | zbl = 0042.38203
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| }}</ref>
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| ==Application==
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| If the inverse of <math>A</math> is already known, the formula provides a
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| [[Computational complexity theory|numerically]] [[Computationally expensive|cheap]] way
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| to compute the inverse of <math>A</math> corrected by the matrix <math>uv^T</math> | |
| (depending on the point of view, the correction may be seen as a
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| [[perturbation theory|perturbation]] or as a [[rank (linear algebra)|rank]]-1 update).
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| The computation is relatively cheap because the inverse of <math>A+uv^T</math>
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| does not have to be computed from scratch (which in general is expensive),
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| but can be computed by correcting (or perturbing) <math>A^{-1}</math>.
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| Using [[unit column]]s (columns from the [[identity matrix]]) for <math>u</math> or <math>v</math>, individual columns or rows of <math>A</math> may be manipulated
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| and a correspondingly updated inverse computed relatively cheaply in this way.<ref>Langville, Amy N.; and Meyer, Carl D.; "Google's PageRank and Beyond: The Science of Search Engine Rankings", Princeton University Press, 2006, p. 156</ref> In the general case, where <math>A^{-1}</math> is a <math>n</math> times <math>n</math> matrix
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| and <math>u</math> and <math>v</math> are arbitrary vectors of dimension <math>n</math>, the whole matrix is updated<ref name="B1951" /> and the computation takes <math>3n^2</math> scalar multiplications.<ref>[http://www.alglib.net/matrixops/general/invupdate.php Update of the inverse matrix by the Sherman–Morrison formula]</ref> If <math>u</math> is a unit column, the computation takes only <math>2n^2</math> scalar multiplications. The same goes if <math>v</math> is a unit column. If both <math>u</math> and <math>v</math> are unit columns, the computation takes only <math>n^2</math> scalar multiplications.
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| ==Verification==
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| We verify the properties of the inverse.
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| A matrix <math>Y</math> (in this case the right-hand side of the Sherman–Morrison formula)
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| is the inverse of a matrix <math>X</math> (in this case <math>A+uv^T</math>)
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| if and only if <math>XY = YX = I</math>.
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| We first verify that the right hand side (<math>Y</math>) satisfies <math>XY = I</math>.
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| :<math>XY = (A + uv^T)\left( A^{-1} - {A^{-1} uv^T A^{-1} \over 1 + v^T A^{-1}u}\right)</math>
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| ::<math>= AA^{-1} + uv^T A^{-1} - {AA^{-1}uv^T A^{-1} + uv^T A^{-1}uv^T A^{-1} \over 1 + v^TA^{-1}u}</math>
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| ::<math>= I + uv^T A^{-1} - {uv^T A^{-1} + uv^T A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}</math>
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| ::<math>= I + uv^T A^{-1} - {u(1 + v^T A^{-1}u) v^T A^{-1} \over 1 + v^T A^{-1}u}</math>
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| Note that <math>v^T A^{-1}u</math> is a scalar, so <math>(1+v^T A^{-1}u)</math> can be factored out, leading to:
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| :<math>XY= I + uv^T A^{-1} - uv^T A^{-1} = I.\,</math>
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| In the same way, it is verified that
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| :<math>YX = \left(A^{-1} - {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}\right)(A + uv^T) = I.</math>
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| Following is an alternate verification of the Sherman–Morrison formula using the easily verifiable identity
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| : <math>( I+wv^T )^{-1}=I-\frac{wv^T}{1+v^Tw}</math>
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| Let <math>u=Aw</math> and <math>A+uv^T=A\left( I+wv^T \right)</math>, then
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| : <math>( A+uv^T )^{-1}=( I+wv^T )^{-1}{A^{-1}}=\left( I-\frac{wv^T}{1+v^Tw} \right)A^{-1}</math>
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| Substituting <math>w={{A}^{-1}}u</math> gives
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| : <math>( A+uv^T )^{-1}=\left( I-\frac{A^{-1}uv^T}{1+v^TA^{-1}u} \right)A^{-1}= {A^{-1}}-\frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}</math>
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| == See also ==
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| * The [[matrix determinant lemma]] performs a rank-1 update to a [[determinant]].
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| * [[Woodbury matrix identity]]
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| * [[Quasi-Newton method]]
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| * [[Binomial inverse theorem]]
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| * [[Bunch–Nielsen–Sorensen formula]]
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| == References ==
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| {{reflist}}
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| == External links ==
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| * {{MathWorld|title=Sherman–Morrison formula|urlname=Sherman-MorrisonFormula}}
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| {{DEFAULTSORT:Sherman-Morrison formula}}
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| [[Category:Linear algebra]]
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| [[Category:Matrix theory]]
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Nice to satisfy you, my name is Refugia. He is truly fond of performing ceramics but he is having difficulties to find time for it. Her family members life in Minnesota. Managing people has been his working day job for a while.
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