Homework #3 -- Math 307/101, Fall 2011, Joel Friedman

PROBLEMS 1 and 2. Do Problem 10 of the Supplemental Exercises, for
Problems 6 and 7.  (I.e., do Problems 6 and 7 for the G matrix instead
of the H matrix.) Your solutions will probably be a function of alpha.
Comment on how this function varies as alpha varies from 1 to 0.

PROBLEM 3. For arbitrary reals p,q, write out the binomial expansion for
(p+q)^4.  Write out a similar expansion for (p-q)^4, and use these two
expressions to write out expansions for

( (p+q)^4 + (p-q)^4 )/ 2 

and 

( (p+q)^4 - (p-q)^4 )/ 2 .

Show that the matrix  A = [ p , q ; q , p ]  has p+q and p-q as eigenvalues.
Write out an expansion for A^4, and explain its relation to the
above expansions involving ( p +/- q )^4 and how diagonalization can be
used to understand this relation.
[If you really don't want to deal with 4th powers, it's OK to do this
with 3rd powers.]  Explain what happens with n-th powers.
[Note: you are essentially tying down some of the details of class discussion
on powers of this A, but you are not assuming that p + q = 1.]