I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda\_i$, then I add another matrix to it as: $A+uv^\top$ where $u$ $(n\times 1)$ and $v$ $(n\times 1)$ are column vector. Also, $A=yy^\top$ with $y$ a $(n-1)$ rank matrix, so $A$ is symmetric.


Is there any way to calculate the new eigenvalues and eigenvectors of $(uv^\top+yy^\top)$ using the information from the vector $u$,$v$ and the eigenvalues and eigenvectors of $yy^\top$？


(The eigenvalues could be calculated by the algorithm given in Golub's paper,but it seems that the Bunch-Nielsen-Sorensen formula is unable to calculate the eigenvectors)



Eigenvectors of a non-symmetric rank-one update of a symmetric matrix
You can still use the [Bunch–Nielsen–Sorensen formula](https://en.wikipedia.org/wiki/Bunch%E2%80%93Nielsen%E2%80%93Sorensen_formula), but you will need to distinguish between left and right eigenvectors. These are different, because the rank-one-update is not symmetric. So if you work in the basis where $A$ is diagonal, with elements $a\_k$ on the diagonal, and the vectors $u$ and $v$ in this basis are $\tilde{u}$ and $\tilde{v}$, the $i$-th left and right eigenvectors at eigenvalue $\lambda\_i$ will have elements $\tilde{u}\_k/(a\_k-\lambda\_i)$ and $\tilde{v}\_k/(a\_k-\lambda\_i)$.


The nonsymmetric BNS formula follows from the nonsymmetric
[Sherman–Morrison formula](https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula), applied to $(A-\lambda I+uv^\top)^{-1}$.


