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, $v{v}^T$, 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 $(A-\tilde{\lambda }I+v{v}^T{)}^{-1}$.

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