Question

Modulus of the roots of the characteristic equation for an orthogonal matrix $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$ is

Answer 1

We have, $\begin{array}{cc}|A-\lambda I|& =\left[\begin{array}{cc}\cos \theta -\lambda & -\sin \theta \\ \sin \theta & \cos \theta -\lambda \end{array}\right]\\ & =(\cos \theta -\lambda {)}^2+{\sin }^2\theta \end{array}$
Therefore characteristic equation of $A$ is $(\cos \theta -\lambda {)}^2+{\sin }^2\theta =0$ $\Rightarrow \cos \theta -\lambda =\pm i\sin \theta$ $\Rightarrow \lambda =\cos \theta \pm i\sin \theta$ which are of unit modulus.