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 [math]\displaystyle{ A }[/math] and the outer product, [math]\displaystyle{ v v^T }[/math], of vector [math]\displaystyle{ v }[/math] with itself.
Statement
Let [math]\displaystyle{ \lambda_i }[/math] denote the eigenvalues of [math]\displaystyle{ A }[/math] and [math]\displaystyle{ \tilde\lambda_i }[/math] denote the eigenvalues of the updated matrix [math]\displaystyle{ \tilde A = A + v v^T }[/math]. In the special case when [math]\displaystyle{ A }[/math] is diagonal, the eigenvectors [math]\displaystyle{ \tilde q_i }[/math] of [math]\displaystyle{ \tilde A }[/math] can be written
- [math]\displaystyle{ (\tilde q_i)_k = \frac{N_i v_k}{\lambda_k - \tilde \lambda_i} }[/math]
where [math]\displaystyle{ N_i }[/math] is a number that makes the vector [math]\displaystyle{ \tilde q_i }[/math] normalized.
Derivation
This formula can be derived from the Sherman–Morrison formula by examining the poles of [math]\displaystyle{ (A-\tilde\lambda I+vv^T)^{-1} }[/math].
Remarks
The eigenvalues of [math]\displaystyle{ \tilde A }[/math] were studied by Golub.[2]
Numerical stability of the computation is studied by Gu and Eisenstat.[3]
See also
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. 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
- Rank-One Modification of the Symmetric Eigenproblem at EUDML
- Some Modified Matrix Eigenvalue Problems
- A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem
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