I am working on a physics project for which I need to use perturbation theory to calculate the first- and second-order corrections to the eigenvalues and eigenvectors of a perturbed matrix. The unperturbed matrix is real and symmetric, and the eigenvalues and eigenvectors are easy to calculate. However, the perturbed matrix is complex and non-Hermitian (the perturbation introduces complex components on the main diagonal). I am new to perturbation theory. My question is whether I can use the standard matrix perturbation theory for Hermitian Hamiltonians, as explained in Chapter 6 of "Introduction to Quantum Mechanics" by Griffiths. Clearly, the unperturbed matrix needs to be Hermitian, but it doesn't seem as if the perturbed one has to be. I would just like to double-check this.
The Attempt at a Solution
I went through the derivation in Griffiths, and didn't see the assumption that the perturbed matrix has to be Hermitian being used anywhere.