Definition (Invariant subspace)

Let $T\in\mathscr{L}(V)$. A subspace $U\subset V$ is called invariant under $T$ if, for every $u\in U$, $T(u)\in U$

Definition (A-invariant)

Let $S\subseteq \mathbb{R}^{n}$ be an $r$-dimensional Subspace. Let $A\in\mathcal{M}_{n}(\mathbb{R})$, we say $S$ is $A$-invariant if $$\{ A\eta \mid \eta \in S \}=:AS\subseteq S$$