Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $$b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}.$$ Let $M \in \mathcal{B}_{n}$, $X \in \mathcal{M}_{n,1}(\mathbf{C})$. Show that the sequence $\left(\left\|M^{k}X\right\|\right)_{k \in \mathbf{N}}$ is bounded. If $\lambda \in \sigma(M)$, if $X$ is an eigenvector of $M$ associated with $\lambda$, express for $k \in \mathbf{N}$, the vector $M^{k}X$ in terms of $\lambda$, $k$ and $X$. Deduce that $\sigma(M) \subset \mathbb{D}$.
Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set
$$b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}.$$
Let $M \in \mathcal{B}_{n}$, $X \in \mathcal{M}_{n,1}(\mathbf{C})$. Show that the sequence $\left(\left\|M^{k}X\right\|\right)_{k \in \mathbf{N}}$ is bounded.\\
If $\lambda \in \sigma(M)$, if $X$ is an eigenvector of $M$ associated with $\lambda$, express for $k \in \mathbf{N}$, the vector $M^{k}X$ in terms of $\lambda$, $k$ and $X$. Deduce that $\sigma(M) \subset \mathbb{D}$.