Bunch–Nielsen–Sorensen formula

In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, 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 $(A-{\tilde {\lambda }}I+vv^{T})^{-1}$.

Remarks

The eigenvalues of ${\tilde {A}}$ were studied by Golub.

Numerical stability of the computation is studied by Gu and Eisenstat.

See also

• Sherman–Morrison formula

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