I need to obtain an approximate expression for the eigenvector corresponding to the largest real eigenvalue of a matrix, as well as the largest eigenvalue. Note that I mean largest not in absolute value, but largest in real value. The matrix can be approximately diagonal in some cases, but not always.
I have tried perturbation theory but it is not working very well. Are there other approximation schemes for the eigenvalues and eigenvectors of a matrix that I can try?