I could not find a complete solution for the next problem:
Let $A \sim B \iff A = PBP^{-1}$ be the equivalence relation for $M_2(R)$ matrices. Here $P$ is an invertible matrix.
Describe the quotient set $X/\!\sim$ and the quotient space? Here $X = \left\lbrace \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right\rbrace$.
Could someone help, please?
I tried to check the trivial case: is this quotient topology discrete? The answer is - No. Counterexample: $$ A_0 = \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}, B_0 = \begin{pmatrix} 0 & 0\\ 1 & 1 \end{pmatrix}, \lambda_{1,2} = \left\lbrace1,0\right\rbrace $$