Finding real numbers $a,b$ and vectors $v,w$ such that a condition is met with complex eigenvalues

Let $u$ be a complex eigenvector corresponding to the eigenvalue of $\lambda$. We have $Au=\lambda u$ and $A\bar u=\bar\lambda\bar u$ (since $A$ is a real matrix). Now let $u=x+iy$ and $\lambda=\alpha+i\beta$, where $x$ and $y$ are real vectors, $\alpha$ and $\beta$ are real numbers. Then \begin{eqnarray*} A(x+iy)&=&Ax+iAy,\\ \lambda u&=&(\alpha x-\beta y)+i(\beta x+\alpha y). \end{eqnarray*} Hence it follows that \begin{eqnarray*} Ax &=& \alpha x-\beta y, \\ Ay &=& \beta x+\alpha y. \end{eqnarray*} Or, using the notation of your teacher \begin{eqnarray*} A(x,y) &=& (x,y)\left(% \begin{array}{cc} \alpha & \beta \\ -\beta & \alpha \\ \end{array}% \right). \end{eqnarray*} Now we get $$ (x,y)^{-1}A(x,y)=\left(% \begin{array}{cc} \alpha & \beta \\ -\beta & \alpha \end{array}% \right). $$