Chapter 8.8, Problem 29E

(a)

To determine

Show that $D=R^{-1}TR$ , where $R=\left[\begin{array}{cc}i& -i\\ 1& 1\end{array}\right]$ and $T=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$

(b)

To determine

Show that $S=P{R}^{-1}$ is a $2\times 2$ matrix whose columns are real and imaginary components of the eigen vectors of $A$ .

(c)

To determine

Show that, we can write $A$ as $A=rS\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]S^{-1}$ , where $r=\sqrt{a^2+b^2}$ is the modulus of the eigen values and $\theta$ is the argument.

(d)

To determine

Show that $A^t=r^{t}S\left[\begin{array}{cc}\cos t\theta & -\sin t\theta \\ \sin t\theta & \cos t\theta \end{array}\right]S^{-1}$