grandes-ecoles 2020 Q29

grandes-ecoles · France · centrale-maths1__pc Matrices Eigenvalue and Characteristic Polynomial Analysis

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$. Show that $MX$, $J_{n} X$ and $J_{n} MX$ are eigenvectors of $M^{2}$ and give the eigenvalues associated with each of these vectors.

Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$. Show that $MX$, $J_{n} X$ and $J_{n} MX$ are eigenvectors of $M^{2}$ and give the eigenvalues associated with each of these vectors.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 Q27 Q28 Q29 Q30 Q31 Q32 Q33 Q34 Q35 Q36