Let $A \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ be a matrix and let $\eta$ be a strictly positive real number. (a) For $x \in \mathbb { C } ^ { 2 }$, show that the series $$\sum _ { n } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ is convergent. We denote $$N ( x ) = \sum _ { n = 0 } ^ { \infty } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ the sum of this series. (b) Show that $x \mapsto N ( x )$ is a norm on $\mathbb { C } ^ { 2 }$, which satisfies the following inequality $$\forall x \in \mathbb { C } ^ { 2 } , \quad N ( A x ) \leqslant ( \rho ( A ) + \eta ) N ( x ) .$$ (c) Show that there exists a real $C > 0$ such that for all $x \in \mathbb { C } ^ { 2 }$ we have $$\| x \| \leqslant N ( x ) \leqslant C \| x \|$$
Let $A \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ be a matrix and let $\eta$ be a strictly positive real number.
(a) For $x \in \mathbb { C } ^ { 2 }$, show that the series
$$\sum _ { n } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$
is convergent.
We denote
$$N ( x ) = \sum _ { n = 0 } ^ { \infty } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$
the sum of this series.
(b) Show that $x \mapsto N ( x )$ is a norm on $\mathbb { C } ^ { 2 }$, which satisfies the following inequality
$$\forall x \in \mathbb { C } ^ { 2 } , \quad N ( A x ) \leqslant ( \rho ( A ) + \eta ) N ( x ) .$$
(c) Show that there exists a real $C > 0$ such that for all $x \in \mathbb { C } ^ { 2 }$ we have
$$\| x \| \leqslant N ( x ) \leqslant C \| x \|$$