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Bunch–Nielsen–Sorensen formula

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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 and the outer product, , of vector with itself.

Statement

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Let denote the eigenvalues of and denote the eigenvalues of the updated matrix . In the special case when is diagonal, the eigenvectors of can be written

where is a number that makes the vector normalized.

Derivation

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This formula can be derived from the Sherman–Morrison formula by examining the poles of .

Remarks

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The eigenvalues of were studied by Golub.[2]

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

See also

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References

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  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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