Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum_{i,j}{a_{ij}}z_i\frac{\partial}{\partial z_j}$ be the linear term of $v$ and suppose that the matrix $a_{ij}$ is invertible and all its eigenvalues have modulus different from $1$. Is it true that for some holomorphic coordinates $w_i$ in a neighbourhood of $0$ we have $v=\sum_{i,j}{a_{ij}}w_i\frac{\partial}{\partial w_j}$?

If yes, where could I find such a statement? If not, what would be a counterexample? I am happy to assume that the eigenvalues of $A$ all have modulus less than $1$.