grandes-ecoles 2019 Q11

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

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$. Assume $\operatorname{Im} u \neq \operatorname{Ker} u$ and that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right), v_1, \ldots, v_{n-2r}\right)$ is a basis of $E$.
What is the matrix of $u$ in this basis?

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$. Assume $\operatorname{Im} u \neq \operatorname{Ker} u$ and that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right), v_1, \ldots, v_{n-2r}\right)$ is a basis of $E$.

What is the matrix of $u$ in this basis?

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