Let $x^2+a_1x+a_0\in\mathbb{R}[x]$, irreducible polynomial. It's roots are $\alpha \pm \beta i$. Show that $A=\left( {\matrix{ 0 & { - {a_0}} \cr 1 & { - {a_1}} \cr } } \right)$ is similar to $B= \left( {\matrix{ \alpha & { \beta} \cr -\beta & { \alpha} \cr } } \right)$
I was guided to find an eigen vector of this matrix. So I looked at $\ker(B-(\alpha+i\beta)I_2)$ and found it's basis is $\left\{ (-i,1) \right\}$.
So we can take $v = -i +1$ as an eigen-vector. Now I was told to look at the real part and the imaginary part and figure out something, but I don't see how is it gonna help here.