How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

Question

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.

Answer 1

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.

Answer 2

A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}(x).$$ The proof only needs the fact that a determinant doesn't change when a multiple of one row is added to a different row, so I don't see why it wouldn't be true for all fields.