I have the following question from a past Algebra qualifying exam:
Let $G=GL(4,\mathbb{C})$ be the group of $4\times4$ invertible matrices with complex entries. List in a precise way the conjugacy classes of $G$.
We call two matrices $A$ and $B$ similar if there exists an invertible matrix $P$ such that $$A=PBP^{-1}.$$ It seems like I'm supposed to be using some result about similar invertible matrices that I don't seem to be familiar with. Can someone point me in the right direction here?