Bunch–Nielsen–Sorensen formula

In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula,[1] named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix $A$ and the outer product, $vv^{T}$ , of vector $v$ with itself.

Statement

Let $\lambda _{i}$ denote the eigenvalues of $A$ and ${\tilde {\lambda }}_{i}$ denote the eigenvalues of the updated matrix ${\tilde {A}}=A+vv^{T}$ . In the special case when $A$ is diagonal, the eigenvectors ${\tilde {q}}_{i}$ of ${\tilde {A}}$ can be written

({\tilde {q}}_{i})_{k}={\frac {N_{i}v_{k}}{\lambda _{k}-{\tilde {\lambda }}_{i}}}

where $N_{i}$ is a number that makes the vector ${\tilde {q}}_{i}$ normalized.

Derivation

This formula can be derived from the Sherman–Morrison formula by examining the poles of $(A-{\tilde {\lambda }}I+vv^{T})^{-1}$ .

Remarks

The eigenvalues of ${\tilde {A}}$ were studied by Golub.[2]

Numerical stability of the computation is studied by Gu and Eisenstadt.[3]

gollark: It could do that, but there's no system to listen for such events.

gollark: Or Embedded HQ9+.

gollark: HQ9+.

gollark: I guess?

gollark: I could make it print those when a PotatOS Incident Report is sent.

References

Bunch, J. R.; Nielsen, C. P.; Sorensen, D. C. (1978). "Rank-one modification of the symmetric eigenproblem". Numerische Mathematik. 31: 31–48. doi:10.1007/BF01396012.
Golub, G. H. (1973). "Some Modified Matrix Eigenvalue Problems". SIAM Review. 15 (2): 318–334. CiteSeerX 10.1.1.454.9868. doi:10.1137/1015032.
Gu, M.; Eisenstat, S. C. (1994). "A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem". SIAM Journal on Matrix Analysis and Applications. 15 (4): 1266. doi:10.1137/S089547989223924X.

External links

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[BNS78-1] Bunch, J. R.; Nielsen, C. P.; Sorensen, D. C. (1978). "Rank-one modification of the symmetric eigenproblem". Numerische Mathematik. 31: 31–48. doi:10.1007/BF01396012.

[GOLUB73-2] Golub, G. H. (1973). "Some Modified Matrix Eigenvalue Problems". SIAM Review. 15 (2): 318–334. CiteSeerX 10.1.1.454.9868. doi:10.1137/1015032.

[GE92-3] Gu, M.; Eisenstat, S. C. (1994). "A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem". SIAM Journal on Matrix Analysis and Applications. 15 (4): 1266. doi:10.1137/S089547989223924X.