grandes-ecoles 2017 Q10

grandes-ecoles · France · x-ens-maths__pc Matrices Matrix Norm, Convergence, and Inequality

Let $A \in M _ { n } ( \mathbb { C } )$. a) Show that, for all $k \in \mathbb { N } ^ { * } , \left\| A ^ { k } \right\| \geqslant \rho ( A ) ^ { k }$. b) We define the subset of $\mathbb { R } _ { + }$ $$E _ { A } = \left\{ \alpha > 0 \left\lvert \, \lim _ { k \rightarrow + \infty } \left( \frac { A } { \alpha } \right) ^ { k } = 0 \right. \right\} .$$ Show that $\left. E _ { A } = \right] \rho ( A ) , + \infty [$.

Let $A \in M _ { n } ( \mathbb { C } )$.\\
a) Show that, for all $k \in \mathbb { N } ^ { * } , \left\| A ^ { k } \right\| \geqslant \rho ( A ) ^ { k }$.\\
b) We define the subset of $\mathbb { R } _ { + }$
$$E _ { A } = \left\{ \alpha > 0 \left\lvert \, \lim _ { k \rightarrow + \infty } \left( \frac { A } { \alpha } \right) ^ { k } = 0 \right. \right\} .$$
Show that $\left. E _ { A } = \right] \rho ( A ) , + \infty [$.

Paper Questions

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