Consider a vector field $S$ on $\mathbb{C}^n$. I need the formal definition of a linear diagonalizable vector field. This means that $S$ can be written under the form $$ S(z)=\sum_{i-1}^{n}\lambda_i z_i\partial z_i $$ where $z=(z_1,...,z_n)\in\mathbb{C}^n$, $\lambda_i\in\mathbb{C}$ for all $i\in\{ 1,...,n \}$ and $\partial z_i$ is the coordinate notation for a vector field. A teacher of mine told me that the formal definition is that there exists a biholomorphism $\phi:\mathbb{C}^n\rightarrow\mathbb{C}^n$ such that the pullback os $S$ under $\phi$ is in the forementioned form, that is,
$$ \phi^{*}S(z)=\sum_{i-1}^{n}\lambda_i z_i\partial z_i $$
Is this correct? Can someone indicate me a reference where I can find such definition? Also, can we refer to the $\lambda_i$ values as the $eigenvalues$ of $S$?
Any help is appreciated.