I am trying to understand how a perturbation of a skew symmetric matrix by another skew symmetric matrix affects the dominant eigenvector corresponding to $\lambda=0$.
Specifically, let $S$ be an $n \times n$ real-valued skew symmetric matrix , with $n$ being odd. Assume $S$ has eigenvectors $(e_1,...,e_n)$ and corresponding eigenvalues $(\lambda_1,...,\lambda_n)$, sorted such that $e_1=0$ (which is guaranteed because $n$ is odd). Thus, $e_1$ is the fixed point solution of $S$.
Let $M$ be another skew symmetric matrix of size $n\times n$, with eigenvalues $(b_1,...,b_n)$ and eigenvalues $(\theta_1,...,\theta_n)$.
For some small $0<\epsilon\ll1$, define:
Then $B$ is likewise skew symmetric, and can be thought of as the perturbation of $S$ by the matrix $M$. Can we approximate or calculate the leading eigenvector of $B$, corresponding to $\lambda=0$, using the eigenvalue distribution of $B$ and $M$? In other words, can we approximate how much the fixed point of $S$ changes when perturbed by $M$?