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.