[Math] Find all possible Jordan Canonical forms

linear algebra

I am working through the problem below

What are the possible Jordan Canonical forms for a matrix $A \in M_n$ with characteristic polynomial $p_A(t)=(t+3)^4(t-4)^2$? Give reason for your answer.

Everything I have found regarding this question has the minimal polynomial given already. Since the minimal polynomial is not given, do I need to consider the various possibilities of the minimal polynomial as below?

  • $(t+3)^4(t-4)^2$
  • $(t+3)^3(t-4)^2$
  • $(t+3)^2(t-4)^2$
  • $(t+3)(t-4)^2$
  • $(t+3)^4(t-4)$
  • $(t+3)^3(t-4)$
  • $(t+3)^2(t-4)$
  • $(t+3)(t-4)$

Then I would need to consider the various Jordan compositions for each? For instance, I believe for minimal polynomial $(t+3)^4(t-4)^2$, we would have $\begin{pmatrix}
-3 & 1 & 0 & 0 & 0 & 0\\
0 & -3 & 1 & 0 & 0 & 0\\
0 & 0 & -3 & 1 & 0 & 0\\
0 & 0 & 0 & -3 & 0 & 0\\
0 & 0 & 0 & 0 & 4 & 1\\
0 & 0 & 0 & 0 & 0 & 4
\end{pmatrix}$.

I am not sure if any of this is correct or not, so any guidance would be much appreciated!

Best Answer

I read the question as asking what forms of the Jordan canonical form, $J$, of the matrix $A$ are possible given that we know that $p_A(t) = (t + 3)^4(t-4)^3$. Assuming I've read it right, the answer I'm suspecting they're looking for is a matrix like the one you've provided above with each element of each Jordan block on the superdiagonal either a $0$ or a $1$ (which then gives $2^{n-2}$ possibilities here). The reason for this is that since $J$ is upper triangular, only the diagonal elements matter when computing the determinant.

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