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 ${\displaystyle A}$ and the outer product, ${\displaystyle vv^{T}}$, of vector ${\displaystyle v}$ with itself.

Statement

Let ${\displaystyle \lambda _{i}}$ denote the eigenvalues of ${\displaystyle A}$ and ${\displaystyle {\tilde {\lambda }}_{i}}$ denote the eigenvalues of the updated matrix ${\displaystyle {\tilde {A}}=A+vv^{T}}$. In the special case when ${\displaystyle A}$ is diagonal, the eigenvectors ${\displaystyle {\tilde {q}}_{i}}$ of ${\displaystyle {\tilde {A}}}$ can be written

${\displaystyle ({\tilde {q}}_{i})_{k}={\frac {N_{i}v_{k}}{\lambda _{k}-{\tilde {\lambda }}_{i}}}}$

where ${\displaystyle N_{i}}$ is a number that makes the vector ${\displaystyle {\tilde {q}}_{i}}$ normalized.

Derivation

This formula can be derived from the Sherman–Morrison formula by examining the poles of ${\displaystyle (A-{\tilde {\lambda }}I+vv^{T})^{-1}}$.

Remarks

The eigenvalues of ${\displaystyle {\tilde {A}}}$ were studied by Golub.^[2]

Numerical stability of the computation is studied by Gu and Eisenstat.^[3]

See also

• Sherman–Morrison formula