Exercise 17, § 10

Let $K$ be a field (commutative or otherwise), $E$ an $n$-dimensional right vector space over $K$ and $u$ an endomorphism of $E$.

(a) Show that if $n > 1$ and $u$ is not a central homothety, there exists a basis of $E$ such that the matrix of $u$ with respect to this basis is of the form $(\alpha_{ij})$ with $\alpha_{ij} = 0$ for $j > i + 1$ and $\sum_{i=1}^{n-1} \alpha_{i, i+1} = 1$ (argue by induction on $n$).

(b) Show that if $n > 2$, there exists a basis of $E$ such that the matrix of $u$ with respect to this basis is of the form $(\alpha_{ij})$ with $\alpha_{ij} = 0$ for $j > i + 1$ and $\sum_{i=1}^{n-1} \alpha_{i, i+1} = 0$. (Argue by induction on $n$ using (a), noting that it amounts to the same to prove the proposition for $u$ or for $u + \gamma 1_E$, where $\gamma$ belongs to the centre of $K$, and studying separately the case where $u$ is of rank 1 or 2.)

(c) Let $A = (\alpha_{ij})$ be a matrix with the properties described in (b). Also let $S$ be the matrix of order $n$ ($\varepsilon_{ij}$) such that $\varepsilon_{ij} = 0$ except when $i = j + 1$, in which case $\varepsilon_{j+1,j} = 1$. Show that there exists a matrix $B = (\beta_{ij})$ of order $n$ such that $A = \lambda E_{nn} + (BS - SB)$ with $\lambda \in \mathbf{K}$. (Take $B$ such that $\beta_{i1} = 0$ for $1 \leq i \leq n$ and $\beta_{ij} = 0$ for $j > i + 2$.)

(d) Suppose that $\mathbf{K}$ is commutative. Deduce from (c) that every endomorphism $u$ of $E$ such that $\operatorname{Tr}(u) = 0$ can be written as $vw - wv$, where $v$ and $w$ are endomorphisms of $E$.