Proposition 19

— Let E be a finite dimensional vector space over a commutative field K, and let $\mathscr{S}$ be a set of commuting triangularisable endomorphisms of E. Then there exists a basis of E with respect to which the matrix of each element u of $\mathscr{S}$ is lower triangular and the matrix of $u_s$ is diagonal, with the same diagonal elements as that of u.

By Cor. 3 of VII, p. 4.5, the set $\mathfrak{T}_s$ of absolutely semi-simple components of elements of $\mathfrak{T}$ consists of diagonalisable elements which commute with one another, so is diagonalisable (VII, p. 41, Prop. 13), the set $\mathfrak{T}_n$ of nilpotent components of elements of $\mathscr{S}$ consists of nilpotent elements which commute with one another, and each element of $\mathfrak{T}_n$ commutes with each element of $\mathfrak{T}_s$. Arguing as in the proof of Prop. 13 (VII, p. 41), we see that there exists a decomposition of E as a direct sum of subspaces $E_i$, which are invariant under $\mathfrak{T}_s$ and $\mathfrak{T}_n$, and such that the restriction of each element of $\mathscr{S}$, to each $E$, is a homothety. Replacing $E$ by each $E$, in turn, we may assume that the elements of $\mathscr{S}$, are homotheties; it is enough to prove that there exists a basis of E with respect to which the elements of $\mathfrak{T}_n$ are represented by lower triangular matrices with zero diagonals; we are thus reduced to the case where $\mathscr{S}$ consists of nilpotent elements.

Now suppose $E \neq 0$, and let F be a nonzero subspace of $E$, invariant under $\mathfrak{T}$, of minimum dimension. Then for each $u \in \mathfrak{T}$, the kernel of the restriction of $u$ to F is nonzero and invariant under $\mathfrak{T}$ (VII, p. 41, Lemma 3); by the choice of F the restriction of $u$ to F is thus zero for all $u \in \mathfrak{T}$. Let $x \in F,\ x \neq 0$; then $u(x) = 0$ for all $u \in \mathscr{S}$; arguing by induction on the dimension of E, we may suppose that there exists a basis $(\bar{e}_1, ..., \bar{e}_{n-1})$ of the quotient $E' = E/Kx$ such that, for all $u \in \mathfrak{T}$, the endomorphism $\bar{u}$ of $E'$ induced by $u$ has a matrix with respect to this basis which is lower triangular with zero diagonal; if $e_i \in E$ projects onto a, for $i = 1, ..., n-1$, then the basis $(e_1, ..., e_{n-1}, x)$ satisfies the required conditions.

Exercises