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| In [[mathematics]], in particular [[linear algebra]], the '''Bunch–Nielsen–Sorensen formula''',<ref name="BNS78">{{cite doi|10.1007/BF01396012 }}</ref> named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a [[symmetric matrix]] <math>A</math> and the [[outer product]], <math>v v^T</math>, of [[vector (mathematics)|vector]] <math>v</math> with itself.
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| ==Statement==
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| Let <math>\lambda_i</math> denote the eigenvalues of <math>A</math> and <math>\tilde\lambda_i</math> denote the eigenvalues of the updated matrix <math>\tilde A = A + v v^T</math>. In the special case when <math>A</math> is diagonal, the eigenvectors <math>\tilde q_i</math> of <math>\tilde A</math> can be written
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| : <math> (\tilde q_i)_k = \frac{N_i v_k}{q_k - \tilde q_i} </math>
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| where <math>N_i</math> is a number that makes the vector <math>\tilde q_i</math> normamlized.
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| ==Derivation==
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| This formula can be derived from the [[Sherman–Morrison formula]] by examining the poles of <math>(A-\tilde\lambda+vv^T)^{-1}</math>.
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| ==Remarks==
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| The eigenvalues of <math>\tilde A</math> were studied by Golub.<ref name="GOLUB73">{{cite doi|10.1137/1015032}}</ref>
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| Numerical stability of the computation is studied by Gu and Eisenstadt.<ref name=GE92>{{cite doi|10.1137/S089547989223924X}}</ref>
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| ==See also==
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| * [[Sherman–Morrison formula]]
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| == References ==
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| {{reflist}}
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| == External links ==
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| * [https://eudml.org/doc/132565 Rank-One Modification of the Symmetric Eigenproblem] at [https://eudml.org/ EUDML]
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| * [http://www.stat.uchicago.edu/~lekheng/courses/309f10/modified.pdf Some Modified Matrix Eigenvalue Problems]
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| * [http://www.cs.yale.edu/publications/techreports/tr916.pdf A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem]
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| {{DEFAULTSORT:Bunch-Nielsen-Sorensen formula}}
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| [[Category:Linear algebra]]
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Latest revision as of 22:41, 12 December 2014
My name is Winfred (24 years old) and my hobbies are Table tennis and Coloring.
my web site Hostgator Voucher