grandes-ecoles 2015 Q3b

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

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$. Let $j$ be an integer, $1 \leqslant j \leqslant n$, and $\mathcal{V}$ be a vector subspace of $\mathbb{R}^{n}$ of dimension $j$. Show that $$\inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle \leqslant m_{j}.$$ (One may use questions $\mathbf{2c}$ and $\mathbf{3a}$, by choosing $\mathcal{U} = \mathcal{W}_{j}$.)

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$. Let $j$ be an integer, $1 \leqslant j \leqslant n$, and $\mathcal{V}$ be a vector subspace of $\mathbb{R}^{n}$ of dimension $j$. Show that
$$\inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle \leqslant m_{j}.$$
(One may use questions $\mathbf{2c}$ and $\mathbf{3a}$, by choosing $\mathcal{U} = \mathcal{W}_{j}$.)

Paper Questions

Q1a Q1b Q1c Q1d Q2a Q2b Q2c Q3a Q3b Q3c Q4a Q4b Q4c Q4d Q5a Q5b Q6a Q6b Q6c Q7a Q7b Q8a Q8b Q8c Q8d Q8e Q9 Q10a Q10b Q10c