Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$. Show that $A^{(m)}$ is similar to $A^{(0)}$.
Let $\left(A^{(m)}\right)_{m \in \mathbb{N}}$ be a sequence in $\mathbf{M}_n$. We assume that for all $m \in \mathbb{N}$, the matrix $A^{(m+1)}$ is similar to $A^{(m)}$.
Show that $A^{(m)}$ is similar to $A^{(0)}$.