Invariant Subspaces

A (right) 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 an invariant subspace may be spanned by a subset of the eigenvectors of $A$, but since some matrices do not have $n$ eigenvectors, there are invariant subspaces that are not spanned by eigenvectors. For example, the space of all possible vectors is clearly invariant, but it is not spanned by the single eigenvector of $A(0)$ in (2.3). This is discussed further in §2.5.4 below.

A left invariant subspace $\cal Y$ of $A$ analogously satisfies $A^*y \in {\cal Y}$ for all $y \in \cal Y$, and may be spanned by left eigenvectors of $A$.