I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. let $v$ and $u$ be a random column vector. i want to know if it is possible to have eigenvalues of matrix $A+uv^T$.

I don't need its eigenvectors, but it is required to have the most precise eigenvalues. We know that $uv^T$ is also a rank one PSD matrix.

Is there a close form to this problem?

${\bf PS:}$ As you see i found the solution, but after implementation of this i can see a very small error in result, it would be appreciated if anyone know why this is happening, because we didn't used any approximation to get the result and it should be exact.