Homework for Friday, Nov 20.  (due in class on Friday).
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[Also, be sure to work on your mini-POW, due Wednesday...]
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71)  Find the trace, the e.v.'s (eigenvalues) and the eigenvectors of the 
following (nxn) matrices:
<pre>
a)  | 1  0 |       b)  | 1  1 |     c)  | 2  2 |     d)  | 3  1 |
    | 0  2 |           | 1  1 |         | 2  2 |         | 1  3 |

e)  | 2  1 |       f)  | 2  1 |     g)  | 2  1 |   
    | 1  2 |           | 1  1 |         | 0  1 |  

h)  | 5  0  0 |       i)  | 3  0  1 |     j)  | 2  2  2 |   k)  | 2  2  2  2 |  
    | 0  2  1 |           | 0  6  0 |         | 2  2  2 |       | 2  2  2  2 |  
    | 0  1  2 |           | 1  0  3 |         | 2  2  2 |       | 2  2  2  2 |  
                                                                | 2  2  2  2 |  
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[Hint for j) and k):  find the eigenvectors first (by guessing).  Then get the
e.v. that belongs to each (by using the definition).  Which is harder, j) or k) ?  why? 
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72) Consider the following mass-spring system (which is similar to a CO2 molecule):
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          _   k    ___   k    _
         |m|zzzzzz| M |zzzzzz|m|
          -        ---        -
a) Calculate the normal modes and their frequencies.  (How many are there?)
Illustrate each mode with a cartoon or picture.
b) Write down the equation for the position of the left-hand mass as a function 
of time if it is struck at t=0 by an identical mass moving to the right at v_0.


73) Calculate the normal mode frequencies, eigenvectors (draw them),
for the symmetric double mass/spring spring system (3 springs), 

          /|  k    _   k    _    k  |/
          /|zzzzzz|m|zzzzzz|m|zzzzzz|/       Symmetric
          /|       -        -       |/
and,
b) Write down the equation for the position of the left-hand mass as a function 
of time if it is struck at t=0 by an identical mass moving to the right at v_0.
(which then disappears)

74) Calculate the normal mode frequencies, eigenvectors (draw them),
for the asymmetric double mass/spring spring system (2 springs) 

          /|  k    _   k    _
          /|zzzzzz|m|zzzzzz|m|               Asymmetric
          /|       -        -

and,
b) Write down the equation for the position of the left-hand mass as a function 
of time if it is struck at t=0 by an identical mass moving to the right at v_0.