I am trying to understand an eigenvector normalization procedure described in an article [1](appendix B).
The problem involves a complex valued matrix $\mathbf{Z}$, of which each elements is a function of the complex variable $s$, for which we want to find the non-trivial solutions of
$\mathbf{Z}(s_n) \mathbf{I}_n=0$
and
$\mathbf{K}_n\mathbf{Z}(s_n)=0$
with $\mathbf{I}_n$ and $\mathbf{K}_n$ the right and left eigenvectors.
Once the $s_n$ are obtained, it is said that the eigenvectors are normalized so that
$\mathbf{K}_n\mathbf{Z}'(s_n)\mathbf{I}_n=1$
and that this normalization ensures that the dyadic product of the eigenvectors matches the pole residue:
$\mathbf{Z}(s)=\frac{\mathbf{I}_n\mathbf{K}_n}{s-s_n}$ in the vicinity of $s_n$.
I am having difficulties linking the normalization step to the final equation.
Can someone explain to me how this works/give me a reference describing the method in more details? Do I have to compute the derivative $\mathbf{Z}'$ of my matrix to conduct this normalization? I know about the residue theorem and Laurent series.
EDIT
After studying this in more details, I am pretty sure that the last equation should read
$\mathbf{Z}^{-1}(s)=\frac{\mathbf{I}_n\mathbf{K}_n}{s-s_n}$
and that there is a typo in the article. In that case, the equation $\mathbf{K}_n\mathbf{Z}'(s_n)\mathbf{I}_n=1$ simply enforces that the coefficient $c_1$ of the Laurent series of $\mathbf{Z}$ around $s_n$ is equal to $1/\mathbf{I}_n\mathbf{K}_n$ so that the expansion of $\mathbf{Z}$ around $s_n$ is
$\mathbf{Z}(s)=\frac{s-s_n}{\mathbf{I}_n\mathbf{K}_n}$ in the vicinity of $s_n$.
and thus
$\mathbf{Z}^{-1}(s)=\frac{\mathbf{I}_n\mathbf{K}_n}{s-s_n}$ in the vicinity of $s_n$.
[1] (Paywall): https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.7.034006 (Free): https://arxiv.org/abs/1610.04980