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, $vv^{T}$, 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 ${\tilde {A}}=A+vv^{T}$. 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