# Rational Canonical form from given minimal and characteristic polynomial

$A$ is a $4\times 4$ matrix over $F$ with characteristic polynomial $(x-1)^4$ and minimal polynomial $(x-1)^2$. What is the rational canonical form of $A$?

My answer was the following: it is one of the following: $$\begin{bmatrix} 0 & -1 & & \\ 1 & 2 & & \\ & & 1 & \\ & & & 1 \end{bmatrix} or \begin{bmatrix} 0 & -1 & & \\ 1 & 2 & & \\ & & 0 & -1\\ & & 1 & 2 \end{bmatrix}.$$ While our teacher finally reached at only second form. I am not satisfied with that answer.

My question is that whether the first matrix here can also be a rational form?

In general, to write rational canonical form of a matrix, I will proceed as follows: let $$m_A(x)=(x-a_1)^{k_1}(x-a_2)^{k_2}\cdots.$$ For each factor $(x-a_i)^{k_i}$ write one block diagonal companion matrix.

If this fills up the matrix size (i.e. if $m_A(x)$ equals characteristic polynomial, then this is required form.

Otherwise, fill up remaining parts (diagonal blocks) by writing companion matrix of factors $(x-a_i)^{l_i}$ where $l_i \leq k_i$.

Is this correct way?

Yes, you have two possible rational canonical forms given the information you have. Both the matrices you wrote have minimal polynomial $(x-1)^2$ and characteristic polynomial $(x-1)^4$.
To justify that $A$ has one of the two possible canonical forms above, let $a_1 \, | \, a_2 \, | \, \ldots \, | \, a_k$ denote the invariant factors of $A$. The highest invariant factor is always the minimal polynomial so $a_k = (x-1)^2$. The characteristic product of the matrix is the product of the invariant factors so we have a priori two options:
$$a_1(x) = (x-1), a_2(x) = (x-1), a_3(x) = (x-1)^2, \\ a_1(x) = (x-1)^2, a_2(x) = (x-1)^2.$$
• Determining (all possible) rational canonical forms I mean the following: suppose characteristic pol. is $(x-1)^6$ and minimal polynomial is $(x-1)^3$. Then possible forms are obtained by putting Companion matrices of size $\leq 3$; the possibilities will be $3+3$, $3+2+1$, $3+1+1+1$. So there will be three possible rational canonical forms when min. pol. is $(x-1)^3$ and char. poly. is $(x-1)^6$. (This situation is almost similar to that in Jordan theory, in which we consider Jordan blocks; in Rational form, we consider Companion blocks. I would like to ensure whether this is correct.) – Beginner Dec 5 '15 at 6:05