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In this paper, we characterize the dynamic of every abelian subgroups$\mathcal{G}$ of GL($n$, $\mathbb{K}$), $\mathbb{K} = \mathbb{R}$ or$\mathbb{C}$. We show that there exists a $\mathcal{G}$-invariant, dense openset $U$ in $\mathbb{K}^{n}$ saturated by minimal orbits with $\mathbb{K}^{n}-U$ a union of at most $n$ $\mathcal{G}$-invariant vectorial subspaces of$\mathbb{K}^{n}$ of dimension $n-1$ or $n-2$ on $\mathbb{K}$. As a consequence,$\mathcal{G}$ has height at most $n$ and in particular it admits a minimal setin $\mathbb{K}^{n}-\{0\}$.Comment: 16 page

Dynamic of Abelian Subgroups of GL(n, $$\mathbb{C}$$ ): A Structure Theorem