Exercise 1.6.1

${\lambda }_1+{\lambda }_2=\mathit{trace}(Q)=2\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃$

The determinant $\left|Q\right|={\cos <!--\; FUNCTION\; APPLICATION\; -->}^2𝜃+{\sin <!--\; FUNCTION\; APPLICATION\; -->}^2𝜃=1$, whereas ${\lambda }_1{\lambda }_2=(\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃+i\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃)(\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃-i\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃)={\cos <!--\; FUNCTION\; APPLICATION\; -->}^2𝜃+{\sin <!--\; FUNCTION\; APPLICATION\; -->}^2𝜃=1$.

${\overline{x}}_1\cdot x_2=(1+i)\cdot (1+i)=1+i^2=0$

$Q^{-1}=\left[\begin{array}{cc}\hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \\ \hfill -\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill \end{array}\right]$

Its eigenvalues are $\frac{1}{{\lambda }_1}$ and $\frac{1}{{\lambda }_2}$