Proposition 13

— Let E be a finite dimensional vector space over a commutative field K, let $\mathscr{S}$ be a set of endomorphisms of E, and let A be the subalgebra of $\mathrm{End}_K(E)$ generated by $\mathscr{S}$ and $\mathrm{Id}_E$. Then the following conditions are equivalent: (i) $\mathscr{S}$ is diagonalisable. (ii) The K-algebra A is diagonalisable. (iii) The elements of $\mathscr{S}$ are diagonalisable and commute with one another.

If $(e_i)$ is a basis of E with respect to which $\mathscr{S}$ is diagonal, then A is contained in the algebra of endomorphisms which are diagonal with respect to this basis, so is also diagonalisable; if A is diagonalisable, then the same argument shows that $\mathscr{S}$ is diagonalisable. This shows the equivalence of (i) and (ii). Since any two diagonal matrices commute, we have (i) $\rightarrow$ (iii), and it remains to prove the converse. Suppose then that the elements of $\mathscr{S}$ are diagonalisable and commute with each other. We will make use of the following lemma: