Let $A \in \mathbf { M } _ { 2 } ( \mathbf { C } )$ be a matrix and let $\epsilon > 0$ be a real number. (a) Show the existence of a real number $\alpha > 0$ such that for every non-negative integer $n$ the absolute values of the coefficients of $A ^ { n }$ are bounded by $\alpha ( \rho ( A ) + \epsilon ) ^ { n }$. (b) Deduce the existence of a real number $\beta > 0$ such that for every non-negative integer $n$ and every $x \in \mathbb { C } ^ { 2 }$ we have $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$
Let $A \in \mathbf { M } _ { 2 } ( \mathbf { C } )$ be a matrix and let $\epsilon > 0$ be a real number.\\
(a) Show the existence of a real number $\alpha > 0$ such that for every non-negative integer $n$ the absolute values of the coefficients of $A ^ { n }$ are bounded by $\alpha ( \rho ( A ) + \epsilon ) ^ { n }$.\\
(b) Deduce the existence of a real number $\beta > 0$ such that for every non-negative integer $n$ and every $x \in \mathbb { C } ^ { 2 }$ we have
$$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$