.
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, \(v v^T \) , 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\( \tilde A = A + v v^T \) . 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 \( (A-\tilde\lambda+vv^T)^{-1} \) .
Remarks
The eigenvalues of \tilde A were studied by Golub.[2]
Numerical stability of the computation is studied by Gu and Eisenstadt.[3]
See also
Sherman–Morrison formula
References
Bunch, J. R.; Nielsen, C. P.; Sorensen, D. C. (1978). "Rank-one modification of the symmetric eigenproblem". Numerische Mathematik 31: 31. doi:10.1007/BF01396012. edit
Golub, G. H. (1973). "Some Modified Matrix Eigenvalue Problems". SIAM Review 15 (2): 318. doi:10.1137/1015032. edit
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. edit
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
