I was studying Linear algebra, mainly group actions on matrices of real entries but I read a statement regarding matrices of complex entries and I don't know why its true:
Almost every $GL_n$($ℂ$)-orbit intersects the set of diagonal matrices in a non empty finite set.
I was having trouble figuring this out for $n$ dimension so i started with simpler case $n$=$2$, If I work on orbit set containing rank two matrices for instance, they definitely intersect with diagonal matrices ( which are invertible) but still the possibilities of the matrices in the intersection is infinite , how come it says finite ? I will be grateful if someone explains this to me , I am still exploring this subject.