Exercise 5, § 5

a) Let $K$ be an algebraically closed field, and let $A$ be a square matrix of order $n$ over $K$, in the form of a diagonal block $\operatorname{diag}(A_i)$ of lower triangular matrices $A_i$, such that the diagonal entries of $A_i$ are all equal to the same element $a$, of $K$ ($1 \leq i \leq r$), and $\alpha_i \neq \alpha_j$ for $i \neq j$. Show that every square matrix $B$ which commutes with $A$ is a diagonal block $\operatorname{diag}(B_i)$, where $B_i$ is the same size as $A_i$ for $1 \leq i \leq r$. b) Let $M$ be a set of commuting square matrices of order $n$ over $K$. Show that by applying the same similarity to all the matrices in $M$, we can make them all into diagonal blocks $\operatorname{diag}(X_i)$ of lower triangular matrices, where the number $r$ of $X_i$'s and the order $m$, of $X_i$ for each $i$, are the same for each matrix in $M$, and where all the eigenvalues of $X_i$ are equal. (Use $a$) to reduce to the case where $r = 1$, then Lemma 3 of VII, p. 41 to show that in this case all the matrices in $M$ have nonzero common eigenvector, and complete the proof by induction on $n$)