Deflating Subspaces

A pair of right and left deflating subspaces $\cal X$ and $\cal Y$ of $A - \lambda B$ have the same dimension and satisfy $Ax \in \cal Y$ and $Bx \in \cal Y$ for all $x \in \cal X$, and furthermore ${\rm span}_{x \in {\cal X}} \{Ax, Bx\}= \cal Y$. We also write this as $A {\cal X} + B {\cal X} = {\cal Y}$. The simplest example is when $\cal X$ is spanned by a single right eigenvector of $A - \lambda B$ and ${\cal Y}$ is spanned by its image under $A$ and/or $B$. More generally a right deflating subspace may be spanned by a subset of right eigenvectors of $A - \lambda B$, and the left deflating subspace by their images. But since some pencils do not have $n$ independent eigenvectors, there are right deflating subspaces that are not spanned by eigenvectors. For example, see the example in §2.5.2.