Invariant Subspaces

An invariant subspace $\cal X$ of $A$ satisfies $Ax \in \cal X$ for all $x \in \cal X$. We also write this as $A {\cal X} \subset {\cal X}$. The simplest example is when $\cal X$ is spanned by a single eigenvector of $A$. More generally any invariant subspace can be spanned by a subset of the eigenvectors of $A$, although the vectors spanning $\cal X$ do not have to be eigenvectors themselves.