I am interested in the eigenvalue spectrum of the following matrix A A=(id+diag random) M Where M is a given matrix with a known eigenvalue spectrum, lets say for simplicity that it is Hermitian. diag random is a diagonal random matrix where each entry is an independent random gaussian number. And id is the identity matrix. For some cases of M, such as M=id this problem is very easy. But if M is more complicated than that there seem to be no clear way to explicitly write down M's spectrum. Any ideas how I could proceeed?