grandes-ecoles 2019 Q27

grandes-ecoles · France · centrale-maths2__psi Matrices Linear Transformation and Endomorphism Properties

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Prove that $\operatorname{Im} u$ is stable under $u$ and that the endomorphism induced by $u$ on $\operatorname{Im} u$ is nilpotent. Specify its nilpotency index.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.

Prove that $\operatorname{Im} u$ is stable under $u$ and that the endomorphism induced by $u$ on $\operatorname{Im} u$ is nilpotent. Specify its nilpotency index.

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