Justify that if $M \in \mathcal{M}_{n}(\mathbf{C})$, the map $$X \in \Sigma_{n} \longmapsto \|MX\|$$ attains its maximum, which we denote by $\|M\|_{\text{op}}$. Establish the two properties $$\begin{gathered}
\forall M \in \mathcal{M}_{n}(\mathbf{C}), \quad \|M\|_{\mathrm{op}} = \max\left\{\frac{\|MX\|}{\|X\|}; X \in \mathcal{M}_{n,1}(\mathbf{C}) \backslash \{0\}\right\}, \\
\forall (M, M^{\prime}) \in \mathcal{M}_{n}(\mathbf{C})^{2}, \quad \|M^{\prime}M\|_{\mathrm{op}} \leq \|M^{\prime}\|_{\mathrm{op}} \|M\|_{\mathrm{op}}.
\end{gathered}$$
Justify that if $M \in \mathcal{M}_{n}(\mathbf{C})$, the map
$$X \in \Sigma_{n} \longmapsto \|MX\|$$
attains its maximum, which we denote by $\|M\|_{\text{op}}$.\\
Establish the two properties
$$\begin{gathered}
\forall M \in \mathcal{M}_{n}(\mathbf{C}), \quad \|M\|_{\mathrm{op}} = \max\left\{\frac{\|MX\|}{\|X\|}; X \in \mathcal{M}_{n,1}(\mathbf{C}) \backslash \{0\}\right\}, \\
\forall (M, M^{\prime}) \in \mathcal{M}_{n}(\mathbf{C})^{2}, \quad \|M^{\prime}M\|_{\mathrm{op}} \leq \|M^{\prime}\|_{\mathrm{op}} \|M\|_{\mathrm{op}}.
\end{gathered}$$