Eigenvalues of sum of two matrices

Question

I have a family of matrices depending on a parameter $\mu$ of the form: $$A(\mu)=A_0+D(\mu),$$ where $D(\mu)$ is a diagonal matrix and $A_0$ does not depend on $\mu.$ The diagonal of $D(\mu)$ is a vector which looks like: $$\left(0, \ldots ,0, \mu, \ldots, \mu,2\mu,\ldots, 2\mu,\ldots \ldots,8\mu,\ldots,8\mu \right).$$ I need to diagonalize $A(\mu)$ for a large range of values of $\mu,$ and since the dimensions of the matrices I am working with is huge, this is expensive. I was thus wondering whether there's a way to simplify the problem knowing that the $\mu-$dependence is only in the diagonal part of the matrix. Since the diagonalization of $D(\mu)$ is trivial for each $\mu,$ is there some way in which just diagonalizing $A_0$ would allow me to compute the eigenvectors and eigenvalues of $A(\mu)$ much more quickly?