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Bunch–Nielsen–Sorensen formula

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

Statement

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

( q ~ i ) k = N i v k λ k − λ ~ i {\displaystyle ({\tilde {q}}_{i})_{k}={\frac {N_{i}v_{k}}{\lambda _{k}-{\tilde {\lambda }}_{i}}}}

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

Derivation

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

Remarks

The eigenvalues of A ~ {\displaystyle {\tilde {A}}} were studied by Golub.

Numerical stability of the computation is studied by Gu and Eisenstat.

See also

References

  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. S2CID 120776348.
  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.
  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.

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