January 21, 2009

PHY 745 - Problem Set #2

This homework is due Friday, January 23, 2009.

Continue reading Chapter 3 in Tinkham. For the following matrices $M$, find the similar transformation $S$ which creates the related diagonal matrix $d$:

$$d=S^{-1} M S,$$

choosing $S$ to be unitary whenever appropriate.

1. In this example, $\theta$ represents a real number.
   
   $$M = \left( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right).$$
   
   

2. In this example, $\theta$ represents a real number.
   
   $$M = \left( \begin{array}{cc} \cos \theta & \sin \theta \\ \sin \theta & \cos \theta \end{array} \right).$$
   
   

3. In this example, you may wish to ask Maple to help.
   
   $$M = \left( \begin{array}{ccc} 1.0 & 3.0 & 1.0 \\ 0.0 & 2.0 & 0.0 \\ 0.0 & 1.0 & 4.0 \end{array} \right).$$
   
   

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