Question refering to the rational form of a matrix.
I'm a bit cunfused by the rational form of the matrix $A=\begin{bmatrix} 1 & 2 & 0 & 4 \\ 4 & 1 & 2 & 0 \\ 0 & 4 & 1 & 2 \\ 2 & 0 & 4 & 1 \\ \end{bmatrix}$,
I know the rational form is made by the companion matrix of the minimal pylinomial in the first block and then other bloks of the companion matrix of the divisors of the minimal polynomial, then, since the minimal polynomial of $A$ is $m_A(x)=(x+5)(x-7)(x^2-2x+5)=x^4-4x^3-26x^2+60x-175$, I think the rational form is
$\begin{bmatrix} 0 & 0 & 0 & 175 \\ 1 & 0 & 0 & -60 \\ 0 & 1 & 0 & 26 \\ 0 & 0 & 1 & 4 \\ \end{bmatrix}$,
but in my class notes I founded the rational form is $\begin{bmatrix} -5 & 0 & 0 & 0 \\ 0 & 7 & 0 & 0 \\ 0 & 0 & 0 & -5 \\ 0 & 0 & 1 & 2 \\ \end{bmatrix}$, which of this is correct and why?