Problem 2, Part 1: Adapted norms We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$. For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$ and $\| x \| _ { 1 } = \sum _ { i = 1 } ^ { d } \left| x _ { i } \right|$. Let $A \in \mathrm { M } _ { d } ( \mathbb { C } )$. Determine a necessary and sufficient condition on $A$ for the map $x \mapsto \| A x \| _ { \infty }$ to define a norm on $\mathbb { C } ^ { d }$.
\textbf{Problem 2, Part 1: Adapted norms}
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$.
For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$ and $\| x \| _ { 1 } = \sum _ { i = 1 } ^ { d } \left| x _ { i } \right|$.
Let $A \in \mathrm { M } _ { d } ( \mathbb { C } )$. Determine a necessary and sufficient condition on $A$ for the map $x \mapsto \| A x \| _ { \infty }$ to define a norm on $\mathbb { C } ^ { d }$.