grandes-ecoles 2019 Q30

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 by induction on $p$ that there exist vectors $x_1, \ldots, x_t$ in $E$ such that $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$. One may apply the induction hypothesis to the endomorphism induced by $u$ on $\operatorname{Im}(u)$.

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

Prove by induction on $p$ that there exist vectors $x_1, \ldots, x_t$ in $E$ such that $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$. One may apply the induction hypothesis to the endomorphism induced by $u$ on $\operatorname{Im}(u)$.

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