Approximate largest (in quantity, not magnitude) eigenvalue and eigenvector of a matrix

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?

• If you have a dominant (hopefully complex) eigenvalue, a good "working horse" is the power method (en.wikipedia.org/wiki/Power_iteration). Besides, in your title "in quantity, not magnitude" is not the best way to encompass at the same time the degree of variability of an eigenvalue (here considered in absolute value) and an eigenvector (defined up to a multiplicative constant). – Jean Marie Sep 21 '17 at 20:02
• @JeanMarie I need an approximate analytical expression, not a numerical algorithm. – becko Sep 21 '17 at 20:25
• Analytical expression with the coefficients of the matrix ? What (small !) size are your matrices ? – Jean Marie Sep 21 '17 at 20:54
• @JeanMarie I meant analytical in the same sense that perturbation theory gives you analytical formulas invovling the elements of the matrix.\ – becko Sep 21 '17 at 21:08
• All right but what is the usual size of your matrices ? – Jean Marie Sep 21 '17 at 21:09