Parametrizing unstable and very unstable manifolds

Existence and uniqueness theorems for unstable manifolds are well-known. Here we prove certain refinements. Let $f\colon(\mathbb C^n,0)\to\mathbb C^n$ be a germ of an analytic diffeomorphism, whose derivative $Df(0)$ has eigenvalues $\lambda_1,\dots,\lambda_n$ such that
$$ |\lambda_1|\geq\dots\geq|\lambda_k|>|\lambda_{k+1}|\geq\dots\geq|\lambda_n|, $$
with $|\lambda_k|>1$.
Then there is a unique $k$-dimensional invariant submanifold whose tangent space is spanned by the generalized eigenvectors associated to the eigenvalues $\lambda_1,\dots,\lambda_k$ , and it depends analytically on $f$.
Further, there is a natural parametrization of this “very unstable manifold,” which can be extended to an analytic map $\mathbb C^k\to\mathbb C^n$ when f is defined on all of $\mathbb C^n$, and is an injective immersion if f is a global diffeomorphism.
We also give the corresponding statements for stable manifolds, which are analogous locally but quite different globally.