Problems for next Monday

For next Monday, do exam 2011 problem 1 except part (d), and compendium problems 1-04, 1-05, 1-08 (might be hard - if so, try 1-01 first), 1-12 (b) and 1-17 (those are as posted under the schedule), in addition to the following two problems.
 

Notes:

1. Problem 1 is a highly exam relevant result. You can use it if asked for definiteness.
   (I showed you this assignment in class, but I merge to one single message.)
    
2. Note the "will not be prioritized" before the last part of problem 2.
    

Problem 1: Let A be a symmetric matrix. The objective of this problem is to show the following equivalences:

• A is positive definite if and only if all eigenvalues > 0
• A is negative definite if and only if all eigenvalues < 0
• A is positive semidefinite if and only if all eigenvalues nonnegative
• A is negative semidefinite if and only if all eigenvalues nonpositive
• A is indefinite if and only if some eigenvalue is > 0 and some is  < 0.

Do the following:

• Explain why the equivalences will follow from the following two problems:
  M = max x'Ax subject to ||x|| = 1
  m =  min x'Ax subject to ||x|| = 1
• Show that M = the largest eigenvalue and m = the smallest.

Problem 2: This involves quite a bit fiddling around with matrix algebra. Not necessarily simple, despite all the hints. Throughout the problem, all matrices are n by n square, all vectors are n-vectors.

• Suppose there is a diagonal matrix D and an invertible matrix V such that A = V D V^-1. 
  Show that if so is the case, then
  • A has n linearly independent eigenvectors,
    and
  • those are the columns {v⁽ⁱ⁾} of V,
    and
  • the respective eigenvalue equals d_ii (main diagonal element of D).
• Suppose furthermore that all d_ii are nonnegative and let p be a positive integer. Put B = V C V^-1,  where C is diagonal with c_ii = d_ii ^1/p  (the pth root). 
  Show that
  • A and B have the same eigenvectors,
    and
  • A = B^p
• It is a fact that a positive semidefinite matrix has a unique positive definite square root.  Indeed, one can use the above construction to define the rth power of A for arbitrary r > 0 (rational first, and then a limiting argument as in the real case).
  The following problem will not be prioritized in class; it finds the positive semidefinite pth root when the eigenvalues of A are distinct. (That assumption can be dropped, but it is hard enough already, perhaps?)
  Show that:
  • if V has the property that V' = V^-1 then B is symmetric
    and
  • if A is positive semidefinite and B is symmetric, then B is positive semidefinite.
• We now proceed to prove that we can choose V symmetric if all the eigenvalues are distinct. Notice that we can always assume ||v⁽ⁱ⁾||=1 by scaling it - assume that from now on.
  Let A be symmetric and let u and w be eigenvectors of A, corresponding to distinct eigenvalues.
  • Calculate u'Aw and w'Au
    and use this to
  • show that u'w = 0.
    Then use this to
  • show that V'V=I if we have scaled the v⁽ⁱ⁾ to ||v⁽ⁱ⁾||=1.

Long, isn't it? Well actually, it isn't watertight until one has shown that a symmetric A has n real eigenvalues [edit: counted with multiplicity - and what happens when they are not all simple]. 

- Nils