grandes-ecoles 2025 Q16c

grandes-ecoles · France · polytechnique-maths-a__mp Matrices Linear Transformation and Endomorphism Properties

Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that $$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$ and verify that these subspaces are stable under $H$.

Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that
$$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$
and verify that these subspaces are stable under $H$.

Paper Questions

Q1 Q2a Q2b Q2c Q3a Q3b Q4 Q5 Q6a Q6b Q6c Q6d Q6e Q6f Q7 Q8a Q8b Q8c Q8d Q9 Q10 Q11a Q11b Q11c Q12a Q12b Q12c Q12d Q12e Q13a Q13b Q14a Q14b Q14c Q15 Q16a Q16b Q16c Q17 Q18a Q18b