I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda\_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector.


and 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 $(xx^\top+yy^\top)$ using the information from the vector $x$ and the eigenvalues and eigenvectors of $yy^\top$？



Eigenvalues of a rank-one update of a symmetric matrix
Given eigenvectors and eigenvalues of the symmetric matrix $A$, you can transform to a basis where $A$ is a diagonal matrix $D$; the vector $x$ in that basis transforms to $\tilde{x}$. Then the problem is that of computing the eigenvalues of a rank-one update $D+\tilde{x}\tilde{x}^\top$ of a diagonal matrix. An efficient algorithm is given in a [paper](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.454.9868&rep=rep1&type=pdf) by Golub, page 325. Once you have the eigenvalues the eigenvectors follow directly from the [Bunch–Nielsen–Sorensen formula.](https://en.wikipedia.org/wiki/Bunch%E2%80%93Nielsen%E2%80%93Sorensen_formula)


