Commuting Matrics with minimal polynomials, Determine JCF Suppose that $A \in M_{5 \times 5}(\mathbb{C})$ such that $(A - 2I)^{5} = 0$. Suppose that $B \in M_{5 \times 5}(\mathbb{C})$ such that the minimal polynomial of $B$ is $x^{3} + x$ and $AB = BA$. What are the possible Jordan canonical forms of $A$? I know that the minimal polynomial of $A$ divides $(x-2)^{5}$. How can I use the fact that $A,B$ commute to determine restrictions on the minimal polynomial of $A$?
 A: $B$ has three eigenvalues $0$ and $\pm i$.  The generalized eigenspaces of $B$ must be invariant under $A$: therefore $A$ must have at least three Jordan blocks.  These could be
of sizes either $2,2,1$ or $3,1,1$ or $2,1,1,1$ or $1,1,1,1,1$.
A: Hint: If the normal form of $A$ consists of $k$ Jordan blocks in total and $B$ commutes with $A$, then $B$ can have at most $k$ distinct eigenvalues. To see this, it helps to note that each eigenvector of $B$ must in turn be an eigenvector of $A$.
A: Since the minimal polynomial of $B$ has no multiple roots, we know that we can find a set of eigenbasis with respect to $B$. Note that there are only two possible combinations of dimensions of eigenspaces of $B$: $1+1+3$ or $1+2+2$.
Since $AB=BA$, then $A$ preserves the eigenspaces of $B$, hence consider the JCF of $A$ restricted to each eigenspace of $B$ and we have the following possible JCFs of $A$:
$$\operatorname{diag}\{2,2,J_3(2)\}, \operatorname{diag}\{2,2,2,J_2(2)\},\operatorname{diag}\{2,2,2,2,2\},\operatorname{diag}\{2,J_2(2),J_2(2)\},$$
where $J_k(2)$ denotes the Jordan block of eigenvalue $2$ of order $k$.
