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 $A^{(m)}$. We further assume that this sequence converges to a diagonal matrix $D$. If $P_m$ denotes the characteristic polynomial of $A^{(m)}$, show that the coefficients of $P_m$ converge to those of the characteristic polynomial of $D$ when $m \rightarrow +\infty$. Deduce that the characteristic polynomial of $D$ is equal to that of $A^{(0)}$.
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 $A^{(m)}$. We further assume that this sequence converges to a diagonal matrix $D$. If $P_m$ denotes the characteristic polynomial of $A^{(m)}$, show that the coefficients of $P_m$ converge to those of the characteristic polynomial of $D$ when $m \rightarrow +\infty$.
Deduce that the characteristic polynomial of $D$ is equal to that of $A^{(0)}$.