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 and the outer product, , of vector with itself.

Statement

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

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

Remarks

The eigenvalues of were studied by Golub.[2]

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

gollark: To make *what* work? What specifically is your aim here?
gollark: The `i` would still be a copy of the one in the list.
gollark: No, it would do the same thing.
gollark: When you loop over the tuple, `i` is *also* a new thing which can't affect the variables in the tuple.
gollark: (not that you can mutate tuples anyway)

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.
  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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.