Classify orbit of $G=GL_2(\mathbb{C}) \times GL_2(\mathbb{C})$ acts on the set $M_2(\mathbb{C})$

Question

The group $G=GL_2(\mathbb{C}) \times GL_2(\mathbb{C})$ acts on the set $M_2(\mathbb{C})$ of $2\times 2$ matrices as follows:-

$(f,g)(x)=fxg^{-1}, f,g \in GL_2(\mathbb{C}), x\in M_2(\mathbb{C})$.

I want to classify the orbit of this action, but when I can not see it ( I tried to take arbitrary $x$ matrix in general but its only will be multiplication of three matrix and do not see any pattern in the resulted matrix). so how to start?

Answer 1

Take the Smith normal form of $x$