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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.

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    $\begingroup$ Are you asking how to check whether two linear subspaces $F,F' \subset \mathbf{R}^n$ with $\dim F + \dim F' = n$ intersect? (Say taking $A = I$ for now.) I don't immediately see what answer you're hoping for. All I can think of is that $F \cap F' = 0$ exactly when $F + F' = \mathbf{R}^n$: two such subspaces would be called complementary. $\endgroup$
    – Leo Moos
    Commented Apr 21, 2021 at 20:03
  • $\begingroup$ @LeoMoos: Thanks for your comment. That is a good point $\endgroup$
    – Adam
    Commented Apr 21, 2021 at 20:14
  • $\begingroup$ Maybe it would help if you will tell us which paper do you mean and which place in it you cannot understand? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 21, 2021 at 20:27
  • $\begingroup$ @მამუკაჯიბლაძე : Thanks for your comment. Please see definition 2.12(twisting). There is no example in the paper. I want to give some examples for myself, but I don't know how to check the twisting property $\endgroup$
    – Adam
    Commented Apr 21, 2021 at 21:09
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    $\begingroup$ The condition in that definition is different from what you are asking. It says nothing about $A$-invariance of $F$ or $F'$. Rather it asks for existence of $\ell$ such that $A^\ell(F)\cap F'=0$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 22, 2021 at 4:05

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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?

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