I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that

$A = D + J$

Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.

When $D$ has identical values, the problem is equivalent to finding the eigenvalues of $J$.

So my question is this:

*If $D$ has non-identical values (specifically, non-identical imaginary components),
is there an elementary way to compute the eigenvalues of $A$ ?*

The problem comes from linearising about the origin of a system of $n$ near identical coupled resonators. $D$ relates to the behaviour of each resonator, $J$ relates to the coupling process.