grandes-ecoles 2019 Q2

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

For any matrix $B \in \mathcal { M } _ { N } ( \mathbb { R } )$, we set $\| B \| = \sup _ { \| x \| = 1 } \| B x \|$.
After justifying the existence of $\| B \|$, show that $B \mapsto \| B \|$ is a norm on $\mathcal { M } _ { N } ( \mathbb { R } )$ satisfying $$\forall x \in \mathbb { R } ^ { N } \quad \| B x \| \leq \| B \| \| x \|$$

For any matrix $B \in \mathcal { M } _ { N } ( \mathbb { R } )$, we set $\| B \| = \sup _ { \| x \| = 1 } \| B x \|$.

After justifying the existence of $\| B \|$, show that $B \mapsto \| B \|$ is a norm on $\mathcal { M } _ { N } ( \mathbb { R } )$ satisfying
$$\forall x \in \mathbb { R } ^ { N } \quad \| B x \| \leq \| B \| \| x \|$$

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