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.


$\textbf{Question}:$ 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.



Intersection Grassmanian planes
For example, if we take any basis $e\_1,e\_2,e\_3,e\_4$ of a 4-dimensional vector space, and if $F$ is the span of $e\_1,e\_2$, so $F\in Gr\_2$ and $F'$ is the span of $e\_3,e\_4$, so $F'\in Gr\_2$, then $F\cap F'$ is the set of vectors which are both of the form $ae\_1+be\_2$ and of the form $ce\_3+de\_4$. Every vector admits a unique representation as a linear combination of basis vectors, so $a=b=c=d=0$.
So $F\cap F'=\{0\}$.


How does this trivial observation not help you?


