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

So my question is this:<br>
*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, $A$ relates to the coupling process.