Given an $n \times n$ matrix $A$ and the $n\times n$ all-ones matrix $J = (1)\_{ij}$, I'm interested in the relation between the eigenvalues and eigenvectors of the matrices $A$ and $A+J$, or more generally $A\_t := A + tJ$.


Is there a nice description of the eigenvalues or eigenvectors of $A\_t$ in terms of those of $A$? If not, what about for $t$ small?


It would be great to have an answer for general coefficient fields, but I would also be interested in the case with $A \in M\_n(\mathbb{C})$ or $A \in M\_n(\mathbb{R})$ with all nonnegative or positive entries, if they have nicer answers.


Thank you.



How are eigenvalues and eigenvectors affected by adding the all-ones matrix?
This is a special case of a *rank one perturbation* or a *rank one update*, and there is plenty of work on such. See the [nice 2010 lecture notes by Andre Ran.](http://www.cs.vu.nl/~ran/LectureBerlijn2010.pdf)


