(a) If $B \in \mathbf { M } _ { \ell } ( \mathbb { C } )$ is diagonalizable, show that there exists a norm $\| \cdot \| _ { B }$ on $\mathbb { C } ^ { \ell }$ such that $\| B x \| _ { B } \leqslant \rho ( B ) \| x \| _ { B }$ for every $x \in \mathbb { C } ^ { \ell }$. Hint: one may verify that if $P \in \mathbf { G L } _ { \ell } ( \mathbb { C } )$, then $x \mapsto \| P x \|$ is a norm on $\mathbb { C } ^ { \ell }$. (b) Show that there exists a matrix $C \in \mathbf { M } _ { 2 } ( \mathbb { C } )$ such that, for every norm $N$ on $\mathbb { C } ^ { 2 }$ there exists $y \in \mathbb { C } ^ { 2 }$ such that $N ( C y ) > \rho ( C ) N ( y )$.
(a) If $B \in \mathbf { M } _ { \ell } ( \mathbb { C } )$ is diagonalizable, show that there exists a norm $\| \cdot \| _ { B }$ on $\mathbb { C } ^ { \ell }$ such that $\| B x \| _ { B } \leqslant \rho ( B ) \| x \| _ { B }$ for every $x \in \mathbb { C } ^ { \ell }$. Hint: one may verify that if $P \in \mathbf { G L } _ { \ell } ( \mathbb { C } )$, then $x \mapsto \| P x \|$ is a norm on $\mathbb { C } ^ { \ell }$.\\
(b) Show that there exists a matrix $C \in \mathbf { M } _ { 2 } ( \mathbb { C } )$ such that, for every norm $N$ on $\mathbb { C } ^ { 2 }$ there exists $y \in \mathbb { C } ^ { 2 }$ such that $N ( C y ) > \rho ( C ) N ( y )$.