5. Deduce the value of $\int _ { 0 } ^ { \infty } \frac { \sin t } { t } \mathrm {~d} t$.

\section*{Problem 2: linear recurrent sequences}
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 $\| x \| _ { 1 } = \sum _ { i = 1 } ^ { d } \left| x _ { i } \right|$. We may use without proof the fact that these define norms on the vector space $\mathbb { C } ^ { d }$.

If $A$ is a matrix in $\mathrm { M } _ { d } ( \mathbb { C } )$ we denote by $\operatorname { Sp } ( A )$ the spectrum of $A$ and we define the spectral radius $\sigma ( A )$ by

$$\sigma ( A ) = \max \{ | \lambda | , \lambda \in \operatorname { Sp } ( A ) \} .$$

\section*{Part 1: Adapted norms}
\begin{enumerate}
  \item 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 $\mathbb { C } ^ { d }$.
  \item Given a matrix $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ we define
\end{enumerate}

$$\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty } .$$

a. Show that this defines a norm on $\mathrm { M } _ { d } ( \mathbb { C } )$ and that there exists $x _ { 0 } \in \mathbb { C } ^ { d }$ such that $\left\| x _ { 0 } \right\| _ { \infty } = 1$ and $\left\| A x _ { 0 } \right\| _ { \infty } = \| A \|$.\\
b. Show that for all $( A , B ) \in \mathrm { M } _ { d } ( \mathbb { C } )$ we have $\| A B \| \leqslant \| A \| \cdot \| B \|$.\\
3. For $1 \leqslant i \leqslant d$ we define $L _ { i } = \left( a _ { i , j } \right) _ { 1 \leqslant j \leqslant d }$ as the $i ^ { \mathrm { th } }$ row vector of $A$. Show that

$$\| A \| = \max _ { 1 \leqslant i \leqslant d } \left\| L _ { i } \right\| _ { 1 } .$$

\begin{enumerate}
  \setcounter{enumi}{3}
  \item a. Let $u \in \mathcal { L } \left( \mathbb { C } ^ { d } \right)$ be an endomorphism of $\mathbb { C } ^ { d }$ and $M = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant d }$ the matrix of $u$ in a basis $\mathcal { B } = \left( e _ { 1 } , \ldots , e _ { d } \right)$. Express the matrix $M ^ { \prime } = \left( m _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant d }$ of $u$ in the basis $\mathcal { B } ^ { \prime } = \left( \alpha _ { 1 } e _ { 1 } , \ldots , \alpha _ { d } e _ { d } \right)$, where the $\alpha _ { i }$ are complex numbers.\\
b. Suppose that $M$ is upper triangular. Show that for all $\varepsilon > 0$ we can choose the $\alpha _ { i }$ such that for $j > i$ we have $\left| m _ { i , j } ^ { \prime } \right| < \varepsilon$.
  \item Let $T = \left( t _ { i , j } \right) _ { 1 \leqslant i , j \leqslant d }$ be an upper triangular matrix. Show that for all $\varepsilon > 0$, there exists a norm $\| \cdot \| ^ { \prime }$ on $\mathbb { C } ^ { d }$ such that for all $x \in \mathbb { C } ^ { d }$ we have
\end{enumerate}

$$\| T x \| ^ { \prime } \leqslant ( \sigma ( T ) + \varepsilon ) \| x \| ^ { \prime }$$

(you may choose $\| \cdot \| ^ { \prime }$ in the form $\| x \| ^ { \prime } = \| P x \| _ { \infty }$ for a suitably chosen matrix $P$)\\