Eigenspaces

An eigenspace $\cal X$ of $A - \lambda B$ satisfies $B^{-1}Ax \in \cal X$ for all $x \in \cal X$. We also write this as $B^{-1} A {\cal X} \subset {\cal X}$, or $A {\cal X} \subset B {\cal X}$. The simplest example is when $\cal X$ is spanned by a single eigenvector of $A - \lambda B$. More generally, an eigenspace can be spanned by a subset of eigenvectors of $A - \lambda B$, although the vectors spanning $\cal X$ do not have to be eigenvectors themselves.