So I know the minimal polynomial $m(x)$ must divide $(x-1)(x^2+1)^2$. Since we are working with $4$ by $4$ real matrices,
$m(x) = (x-1)$ or $m(x) = x^2 + 1$ or $m(x) = (x^2+1)^2$ or $m(x) = (x-1)(x^2+1)$.
Do I just do a case-by-case analysis or is there a way to determine which of the four is actually the minimal polynomial?
Also, if it is case-by-case then we would have:
(I) $m(x) = (x-1)$ corresponds to $A$ being the $4$ by $4$ Identity matrix
(II) $m(x) = (x^2+1)^2$ corresponds to the companion matrix of $x^4 + 2x^2 + 1$
(III) $m(x) = (x^2+1)$ means we have two blocks each with companion matrix corresponding to $x^2+1$
(IV) $m(x) = (x-1)(x^2+1)$ means we have one block with companion matrix $x^3 - x^2 + x -1$ and another block corresponding to $x-1$.
Am I on the right track? Is this solution correct?