grandes-ecoles 2025 Q16

grandes-ecoles · France · mines-ponts-maths1__psi Matrices Diagonalizability and Similarity

For every $\lambda \in \mathbf{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, and $N'$ is the matrix such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$. Calculate $(N')^n$ and deduce that $J_n(\lambda)^{-1}$ is similar to $J_n\left(\frac{1}{\lambda}\right)$.

For every $\lambda \in \mathbf{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, and $N'$ is the matrix such that $J_n(\lambda)^{-1} = \frac{1}{\lambda} I_n + N'$.\\
Calculate $(N')^n$ and deduce that $J_n(\lambda)^{-1}$ is similar to $J_n\left(\frac{1}{\lambda}\right)$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21