I need some assistance with the proof for part (b) of the following problem statement:
Problem Statement: Decompose the set $\mathbb{C}^{2\times2}$ of $2\times2$ complex matrices into orbits for the following operations of $GL_{2}(\mathbb{C})$:
(a) left multiplication
(b) conjugation
So, I was able to decompose $\mathbb{C}^{2\times2}$ into four orbits for part (a) fairly simply, but I am a little unsure how to go about part (b).
So I know that in the general case, the orbit of $A\in\mathbb{C}^{2\times2}$ is defined as: $$O_{A}=\left\{B\in\mathbb{C}^{2\times2} \mid B=P^{-1}AP \text{ for } P\in GL_{2}(\mathbb{C})\right\}$$
I was thinking using the fact that for any $A\in\mathbb{C}^{2\times2}$ there is some $P\in GL_{2}(\mathbb{C})$ such that $B=P^{-1}AP$ has Jordan block form, being either a diagonal or triangular matrix? But this would not necessarily be true $\forall A\in\mathbb{C}^{2\times2}, P\in GL_{2}(\mathbb{C})$? Because any $A\in\mathbb{C}^{2\times2}$ would have a distinct $P\in GL_{2}(\mathbb{C})$ such that conjugation by $P$ results in the Jordan form of $A$?
Would the use of an equivalence relation defined as $A\sim B$ iff $B=P^{-1}AP$ for some $P\in GL_{2}(\mathbb{C})$ be helpful?