Hello everyone what I'm trying to solve is:
Let $A$ be a real 4x4 matrix with characteristic polynomial $(x^2 + 3) (x^2 + 1)$. Show that it is similar, in $\Bbb R^4$ to this matrix: $$\begin{pmatrix} 0 & -\sqrt{3} & 0 & 0 \\ \sqrt 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0\end{pmatrix}$$
What I've tried so far:
I've managed to prove they got the same characteristic polynomial and that they are diagonalizable considered as matrices on the complex field, and that they are similar to the same diagonal matrix, therefore the 2 matrice are similar on $\Bbb C$. I've also proved that this implies that they are similar on $\Bbb R$ but I don't understand how to generalize this concept. The excercise tells me also to generalize the concept to a matrix whose characteristic polynomial is product of irreducibile second grade terms, all distinct, and I don't know how to go on it.
thanks to everyone who will to help me. I really appreciate it.