Suppose $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $B: \mathbb{R}^n \rightarrow \mathbb{R}^m$ are linear maps. Let $X \subset \mathbb{R}^n$ be such a linear subspace of $\mathbb{R}^n$ that $A(X)=B(X)$. Where can I read more about such subspaces $X$? Is there a common name for them? I am interested in some sort of a general theory.
Motivation:
If $n=m$ and $B=\operatorname{Id}$ then $X$ is called an A-invariant subspace of $\mathbb{R}^n$. Furthermore, if $\operatorname{dim}(X)=1$ then $X$ is simply spanned by some eigenvector of $A$.
This makes me think of eigenvectors of some matrix as of its "similarity" to the identity matrix, because if $x$ is an eigenvector of $A$ and $X=\operatorname{Span}(x)$ then $A(X)=\operatorname{Id}(X)$. But what if I want to capture the similarity between matrix $A$ and some nonidentity matrix $B$? And what if they are not square? This leads me to the question above.