View source for Bunch–Nielsen–Sorensen formula

In [[Mathematics|mathematics]], in particular [[Linear algebra|linear algebra]], the '''Bunch–Nielsen–Sorensen formula''',<ref name="BNS78">{{Cite journal | last1 = Bunch | first1 = J. R. | last2 = Nielsen | first2 = C. P. | last3 = Sorensen | first3 = D. C. | title = Rank-one modification of the symmetric eigenproblem | doi = 10.1007/BF01396012 | journal = Numerische Mathematik | volume = 31 | pages = 31–48 | year = 1978 | s2cid = 120776348 }}</ref> named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a [[Symmetric matrix|symmetric matrix]] <math>A</math> and the [[Outer product|outer product]], <math>v v^T</math>, of vector <math>v</math> with itself.
==Statement==
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

: <math> (\tilde q_i)_k = \frac{N_i v_k}{\lambda_k - \tilde \lambda_i} </math>

where <math>N_i</math> is a number that makes the vector <math>\tilde q_i</math> normalized.

==Derivation==
This formula can be derived from the [[Sherman–Morrison formula]] by examining the poles of <math>(A-\tilde\lambda I+vv^T)^{-1}</math>.

==Remarks==

The eigenvalues of <math>\tilde A</math> were studied by Golub.<ref name="GOLUB73">{{Cite journal | last1 = Golub | first1 = G. H. | title = Some Modified Matrix Eigenvalue Problems | doi = 10.1137/1015032 | journal = SIAM Review | volume = 15 | issue = 2 | pages = 318–334 | year = 1973 | citeseerx = 10.1.1.454.9868 }}</ref>

Numerical stability of the computation is studied by Gu and Eisenstat.<ref name=GE92>{{Cite journal | last1 = Gu | first1 = M. | last2 = Eisenstat | first2 = S. C. | doi = 10.1137/S089547989223924X | title = A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem | journal = SIAM Journal on Matrix Analysis and Applications | volume = 15 | issue = 4 | pages = 1266 | year = 1994 }}</ref>

==See also==
* [[Sherman–Morrison formula]]

== References ==
{{reflist}}

== External links ==
* [https://eudml.org/doc/132565 Rank-One Modification of the Symmetric Eigenproblem] at [https://eudml.org/ EUDML]
* [http://www.stat.uchicago.edu/~lekheng/courses/309f10/modified.pdf Some Modified Matrix Eigenvalue Problems]
* [http://www.cs.yale.edu/publications/techreports/tr916.pdf A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem]

{{DEFAULTSORT:Bunch-Nielsen-Sorensen formula}}
[[Category:Linear algebra]]

{{Sourceattribution|Bunch–Nielsen–Sorensen formula}}