grandes-ecoles 2016 Q1

grandes-ecoles · France · x-ens-maths__psi Proof Direct Proof of an Inequality

Let $x , y$ be strictly positive vectors of $\mathbb { R } ^ { n }$ and let $S , R$ be two sign diagonal matrices.
(a) Show that $$( S x \mid R y ) \leq ( x \mid y )$$ with equality if and only if $R = S$.
(b) Prove the uniqueness of $S$ in Broyden's theorem.
(c) Show that $$\| S x + R y \| \leq \| x + y \|$$ with equality if and only if $R = S$.

Let $x , y$ be strictly positive vectors of $\mathbb { R } ^ { n }$ and let $S , R$ be two sign diagonal matrices.\\
(a) Show that
$$( S x \mid R y ) \leq ( x \mid y )$$
with equality if and only if $R = S$.\\
(b) Prove the uniqueness of $S$ in Broyden's theorem.\\
(c) Show that
$$\| S x + R y \| \leq \| x + y \|$$
with equality if and only if $R = S$.

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