Finding all rational canonical forms of a matrix in $M_{5}(\mathbb{Q})$ given it's minimum polynomial? Given a $5\times 5$ matrix $A$ over $\mathbb{Q}$, with minimum polynomial $p(x)=(x-2)(x^{2}+1)^{2}$, how do I find all possible rational canonical forms of A? How do I find how many similarity classes exist with the same minimum polynomial? How many Jordan canonical forms of A exist if instead we take the matrix to have entries in $\mathbb{C}$? And in that case, how many similarity classes exist?
 A: In general, a good tactic is to think about the possible Smith Normal Forms, which are in bijection with similarity classes. The last polynomial in the bunch is the minimal polynomial, and the product of all the $\alpha_i$ is the characteristic polynomial. 
In this case, the minimal polynomial $p(X)$is of degree $5$, which is the same degree as the characteristic polynomial, so there is only one possible Smith Normal Form, given by $1,1,1,1, p(x)$. So there is only one possible similarity class of matrices satisfying the condition you state (over $\mathbb{Q}$ or over $\mathbb{C}$.
A: We are given that $A\in M_{5\times 5}(\Bbb Q)$ and that
\begin{align*}                                                                                 
\textstyle{\min_A}(t)                                                                                      
&= (t-2)\left(t^2+1\right)^2 \\                                                                
&= t^5-2\,t^4+2\,t^3-4\,t^2+t-2                                                                
\end{align*}
Since $\deg\min_A(t)=5$  it follows that
$\min_A(t)=\DeclareMathOperator{char}{char}\char_A(t)$.   Thus the rational
canonical form of $A$ is simply    the companion matrix of $\min_A(t)$
$$                                                                                             
\begin{bmatrix}                                                                                
0&0&0&0&2\\                                                                                    
1&0&0&0&-1 \\                                                                                  
0&1&0&0&4\\                                                                                    
0&0&1&0&-2\\                                                                                   
0&0&0&1&2                                                                                      
\end{bmatrix}                                                                                  
$$
To compute the possible Jordan forms $J$ of $A$, note that the eigenvalues of $A$ are
\begin{align*}
\lambda_1 &= 2 & \lambda_2 &= i & \lambda_3 &= -i
\end{align*}
and that the algebraic multiplicities of the eigenvalues are
\begin{align*}
\DeclareMathOperator{amult}{amult}\amult(\lambda_1)
 &=1 & 
\amult(\lambda_2) &= 2 &
\amult(\lambda_3) &= 2
\end{align*}
Since $\min_A(t)=\char_A(t)$ we know that the sizes of the largest Jordan block corresponding to $\lambda_1$, $\lambda_2$, and $\lambda_3$ are $1$, $2$, and $2$ respectively. What does this say about $J$?
