Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. For all $x \in \mathbb { R } ^ { N }$, we set $\| x \| _ { A } = \langle x , A x \rangle ^ { 1 / 2 }$. a) Show that the map $x \mapsto \| x \| _ { A }$ is a norm on $\mathbb { R } ^ { N }$. b) Show that there exist constants $C _ { 1 }$ and $C _ { 2 }$ strictly positive, which we will express in terms of the eigenvalues of $A$, such that $$\forall x \in \mathbb { R } ^ { N } \quad C _ { 1 } \| x \| \leq \| x \| _ { A } \leq C _ { 2 } \| x \| .$$
Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. For all $x \in \mathbb { R } ^ { N }$, we set $\| x \| _ { A } = \langle x , A x \rangle ^ { 1 / 2 }$.
a) Show that the map $x \mapsto \| x \| _ { A }$ is a norm on $\mathbb { R } ^ { N }$.
b) Show that there exist constants $C _ { 1 }$ and $C _ { 2 }$ strictly positive, which we will express in terms of the eigenvalues of $A$, such that
$$\forall x \in \mathbb { R } ^ { N } \quad C _ { 1 } \| x \| \leq \| x \| _ { A } \leq C _ { 2 } \| x \| .$$