Show that the complex matrices $A$ and $B$ are similar.
$A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0& 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ \end{bmatrix} $, $B=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0& -1 & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & 0 & 0 & -i \\ \end{bmatrix} $
My attempt:
The characteristic polynomial of $A$ is $p_A(x)=x^4-1$ and so, $p_A(x)=(x-1)(x+1)(x-i)(x+i)$ then of minimal polynomial is $m_A(x)=(x-1)(x+1)(x-i)(x+i)$ what means $A$ is diagonalizable and $B$ is the matrix diagonal that we obtain by diagonalization, then $A$ is similar to $B$. Is this correct?