grandes-ecoles 2025 Q16

grandes-ecoles · France · mines-ponts-maths1__pc Matrices Diagonalizability and Similarity

For every $\lambda \in \mathbb{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise, 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 $\lambda \in \mathbb{C}$ nonzero, we set $J_n(\lambda) = \lambda I_n + N$, where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise, 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 $J_n\left(\frac{1}{\lambda}\right)$.

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