\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. 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 $\sigma ( A ) = \max \{ | \lambda | , \lambda \in \operatorname { Sp } ( A ) \}$. We define $\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty }$.

Application: norm and spectral radius.\\
a. Let $T \in \mathrm { M } _ { d } ( \mathbb { C } )$ be an upper triangular matrix. Show that for all $\varepsilon > 0$, there exists a constant $C$ such that for all $n$ we have $\left\| T ^ { n } \right\| \leqslant C ( \sigma ( T ) + \varepsilon ) ^ { n }$.\\
b. Show that $\lim _ { n \rightarrow \infty } \left\| T ^ { n } \right\| ^ { 1 / n } = \sigma ( T )$.\\
c. Let now $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ be an arbitrary matrix. Show that $\lim _ { n \rightarrow \infty } \left\| A ^ { n } \right\| ^ { 1 / n } = \sigma ( A )$.\\
d. Show the equivalence
$$A ^ { n } \underset { n \rightarrow \infty } { \longrightarrow } 0 \Leftrightarrow \sigma ( A ) < 1 .$$