I am reading a paper that I don't understand some parts of it.
Let $A$ be a $4\times 4$ matrix, which its char poly is irreducible. Then, the paper said we need to check "for or every pair of $A$-invariant, $F \in Gr(k)$ and $F^{\prime} \in Gr(4-k)$, $A(F) \cap F^{'}=\{0\},"$ which $Gr(k)$ is a Grassmanian of $k$-planes of $\mathbb{R}^4$.
My attempt and questions:
Since the char poly is irreducible, any invariant subspaces are the span of eigenvectors. Let us check the above equation for the eigenspace $H \in Gr(1)$ and $H^{'} \in Gr(3),$ which is the span of three eigenvectors. $A(H)$ is a $4 \times 1$ matrix, but $H^{'}$ is a $3 \times 4$ matrix, what does it mean about their intersection?
I would really appreciate it if one give an example for it.
Thanks in advance.