In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. Let $f$ be a nilpotent endomorphism of $E$. Show that there exists an orthonormal basis of $E$ in which the matrix of $f$ is lower triangular.
In this part, we assume that $\mathbb{K} = \mathbb{R}$ and that $E$ is a Euclidean space. Let $f$ be a nilpotent endomorphism of $E$. Show that there exists an orthonormal basis of $E$ in which the matrix of $f$ is lower triangular.