I came across a problem recently in my linear algebra studies that went something like this:
Let $A$ be a linear transformation on a finite-dimensional space $V$ with characteristic polynomial $(x - 1)^6(x + 6)^2(x - 9)$ and minimal polynomial $m_A(x) = (x - 1)(x + 6)(x -9)$. Let $B$ be a linear transformation which commutes with $A$ and has characteristic polynomial $x^3(x - 2)^6$. Describe all possible minimal polynomials of $B$.
Now, via Cayley-Hamilton, I know that $m_B(x)$ must divide $x^3(x - 2)^6$, so $m_B(x) = x^a(x - 2)^b$, where the possibilities for $a$ are $1, 2, 3$ and for $b$ are $1, 2, 3, 4, 5, 6$, giving us a total of $18$ initial possibilities for the minimal polynomial. Moreover, I know that $A$ and $B$ are simultaneously triangulable, since they commute and operate on a finite-dimensional space $V$. Furthermore, I know $A$ is diagonalizable, since its minimal polynomial has only simple roots; and $A$ is invertible, since none of its eigenvalues is zero.
I would like to straightforwardly limit my choices for $m_B(x)$, but I cannot see a simple approach to eliminating any choices for $a$ and $b$. I tried to see what would happen if $a = b = 1$, but I got nowhere with this. Any ideas as to how I should limit my choices for $a$ and $b$?