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

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

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