List of $2 \times 2$-matrices similar to themselves regardless of the choice of $P$. It is trivial to see that the identity matrix is similar to itself, i.e $I = P^{-1}IP$ for any $P$. Is there a way to determine whether other matrices meet this criteria?
 A: We have
$$A=P^{-1}AP\iff PA=AP$$
which means that $A$ commutes with every invertible matrix $P$. But since the map
$$\mathcal M_n(\Bbb R)\rightarrow \mathcal M_n(\Bbb R), P\mapsto PA -AP$$
is continuous (since it's linear in finite dimensional space) and since $\mathcal{GL}_n(\Bbb R) $ is dense in $\mathcal M_n(\Bbb R)$ then we see that $A$ commutes with all matrix in $\mathcal M_n(\Bbb R)$ hence 
$$A=k I_n$$
The converse is trivial. Conclude.
A: It must be of the form $\lambda I$ for some $\lambda$. Similitude can be seen as base change and the matrix considered as a representation of an endomorphism. Suppose we are in $\mathbb{C}$, so there is at least an eigenvalue $\lambda$ and an eigenvector $v$. Suppose the matrix $A$ has our property. Then $Av= \lambda v$. Then since $P$ is any invertible matrix, you can choose any vector $w$ and find $P$ such that $Pw=v$ Then $Aw=P^{1}APw=P^{-1}Av=\lambda P^{-1}v=\lambda w$. Hence any $w$ is eigenvector with respect to $\lambda$. Hence $A=\lambda I$.
