grandes-ecoles 2019 Q38

grandes-ecoles · France · centrale-maths2__psi Matrices Determinant and Rank Computation

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For every integer $j$ between 1 and $n$, express the number of blocks $J_{\alpha_i}$ of size exactly equal to $j$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.

For every integer $j$ between 1 and $n$, express the number of blocks $J_{\alpha_i}$ of size exactly equal to $j$.

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