# Perturbation of eigenvalues of some special matrices

In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, we want to know if the corresponding path of eigenvalues is holomorphic (as a function of $x$) in a neighborhood of $x=0$. There are some general results, on radii of convergence and whatnot. Now if $A$ is symmetric then simpler and sometimes stronger results exist, and this is developed (for instance) in Kato's classic book on the theory.

Suppose $A=(a_{ij})$ is not quite symmetric, but instead $a_{ii}=0$ and $a_{ji}=1/a_{ij}$ for non-zero $a_{ij}$. Are there certain general results that are simplified with this assumption?

• I've quickly attempted an interpretation of the OP by adding some background, but this can be made much more precise by giving explicit examples of those "general results". – Chris Gerig Jun 7 '15 at 6:31