grandes-ecoles 2015 Q5b

grandes-ecoles · France · x-ens-maths__pc Proof Proof That a Map Has a Specific Property

We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Show that the first component $s_{1}^{\downarrow}$ of $s^{\downarrow}$ is of class $\mathscr{C}^{1}$ on $\mathcal{S}_{2}^{\dagger}(\mathbb{R})$, but not on $\mathcal{S}_{2}(\mathbb{R})$. (One may use question 1d.)

We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Show that the first component $s_{1}^{\downarrow}$ of $s^{\downarrow}$ is of class $\mathscr{C}^{1}$ on $\mathcal{S}_{2}^{\dagger}(\mathbb{R})$, but not on $\mathcal{S}_{2}(\mathbb{R})$. (One may use question 1d.)

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