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

1. 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.
2. Golub, G. H. (1973). "Some Modified Matrix Eigenvalue Problems". SIAM Review 15 (2): 318. 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.

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