grandes-ecoles 2019 Q32

grandes-ecoles · France · centrale-maths2__psi Matrices Diagonalizability and Similarity

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$.
Show that there exist a partition $\sigma = \left(\alpha_1, \ldots, \alpha_k\right)$ of $n$ and a basis $\mathcal{B}$ of $E$ in which the matrix of $u$ is equal to the matrix $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$.

Show that there exist a partition $\sigma = \left(\alpha_1, \ldots, \alpha_k\right)$ of $n$ and a basis $\mathcal{B}$ of $E$ in which the matrix of $u$ is equal to the matrix $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$.

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