Exercise 17, § 10

Let K be a field (commutative or not), E a right vector space of dimension n over K, u an endomorphism of E. a) Now prove that if $n > 1$ and if 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 $\sum_{i=1}^{n-1} \alpha_{i, i+1} = 1$ (reason by induction on n). b) Now prove 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 $\sum_{i=1}^{n-1} \alpha_{i, i+1} = 0$. (Reason by induction on n using a), noting that it amounts to the same thing to prove the proposition for u or for $u + \gamma 1_E$, where $\gamma$ belongs to the center of K, and by studying separately the case where u is of rank 1 or 2.) c) Let $A = (\alpha_{ij})$ be a matrix having the properties described in b). Let, on the other hand, S be the matrix $(\varepsilon_{ij})$ of order n such that $\varepsilon_{ij} = 0$, except for $i = j + 1$ where $\varepsilon_{j+1, j} = 1$. Now prove that there exists a matrix $B = (\beta_{ij})$ of order n such that $A = \lambda E_{nn} + (BS - SB)$ with $\lambda \in K$. (Take B such that $\beta_{i1} = 0$ for $1 \leq i \leq n$ and $\beta_{ij} = 0$ for $j > i + 2$). d) Suppose K commutative. Deduce from c) that every endomorphism u of E such that $\mathrm{Tr}(u) = 0$ can be written $vw - wv$, where v and w are endomorphisms of E.