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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.

Susan Blackford 2000-11-20