Given two algebra automorphisms $\phi, \psi: M_n(\mathbb{R}) \to M_n(\mathbb{R})$ of the real algebra $M_n(\mathbb{R})$, we say they are similar if there exists an algebra automorphism $\alpha: M_n(\mathbb{R}) \to M_n(\mathbb{R})$ such that $\phi = \alpha^{-1}\psi\alpha$, or equivalently, $\alpha \phi = \psi \alpha$.
We say that a map $\xi: A \to A$ is an anti-automorphism of an algebra $A$ if it's a linear automorphism of $A$ and for any two elements $a,b \in A$, $\xi(ab) = \xi(b)\xi(a)$.
I'm interested in whether or not there is a general approach to determining if two algebra automorphisms or anti-automorphisms are similar.
Specifically, I'm trying to make my way through Ian R. Porteous's 'Clifford Algebras and the Classical Groups' and problem 2.4 from chapter 2 asks the following:
Prove that:
$\begin{pmatrix} a & b \\ c & d \end{pmatrix}\mapsto \begin{pmatrix} a & -c \\ -b & d \end{pmatrix}$
and
$\begin{pmatrix} a & b\\ c & d \end{pmatrix} \mapsto \begin{pmatrix} d & b \\ c & a \end{pmatrix}$
are similar anti-automorphisms of $M_2(\mathbb{R})$.