Suppose $k$ is a field, and $m<n$ are nonnegative integers. Let $\operatorname{Gr}(m,n)$ be the Grassmannian (whose points are $m$-dimensional subspaces of a $n$-dimensional linear space). Then we have a natural action: $\operatorname {GL}(n)\times \operatorname{Gr}(m,n)\to \operatorname{Gr}(m,n)$. I want to ask is this action Zariski closed?
Actually, I want to understand: If $A$ is a $n\times n$ matrix, and $Z(A)$ denotes the invertible matrices that are commutative with $A$, for any point $h\in \operatorname{Gr}(m,n)$, is $Z(A)h$ Zariski closed in $\operatorname{Gr}(m,n)$? If so, what functions cut out this closed set?