I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and Grassmanian of $k$-planes, which I don't understand. I know how to check the intersection a plane and a line, but I don't know how to check the intersection Grassmanian of $n-k$-planes and Grassmanian of $k$-planes.

Let me ask my question better. Let $A$ be $n \times n$ matrix. For every pair of $A$-invariant, $F \in Gr(k)$ and $F^{\prime} \in Gr(n-k)$, how I can check whether $F \cap F^{'}=\{0\}?$

I would appreciate it if one could either explain it or introduce some reference.