Note: IML notation is used for matrices in all cases.

1. Consider the matrix A={1 2, 2 -2}. Find the eigenvalues and
   eigenvectors of A.

   Soln:

   The chracteristic equation is:
         lambda^2 + lambda - 4
   Hence the eigenvalues are lambda(1)=2 and lambda(2)=-3 and  
   the eigenvectors are x={2, 1} for lambda=2 and x={1, -2} for 
   lambda=-3.

2. Consider the matrix A={9 -2, -2 6}. answer the following:
   a) Is A symmetric.
   b) Find the eigenvalues and eigenvectors of A.

   Soln:

   a) Yes, since A=A`
   b) The chracteristic equation is:
         lambda^2 - 15*lambda + 50
      Hence the eigenvalues are lambda(1)=10 and lambda(2)=5 and  
      the eigenvectors are x={1, -2} for lambda=10 and x={1, 2} for 
      lambda=5.

3. Consider the matrix A={3 1 1, -1 3 1}. Find the eigenvalues
   and eigenvectors of Z=AA` (i.e. the product of A and its 
   transpose.

   Soln:

   Z = {11 1, 1 11} and the characteristic equation is:
         lambda^2 - 22*lambda + 120
   Hence the eigenvalues are lambda(1)=12 and lambda(2)=10 and  
   the eigenvectors are x={1, 1} for lambda=12 and x={1, -1} for 
   lambda=10.
 
4. Consider the covariance matrix sigma={4 0 0, 0 9 0, 0 0 1}. Find 
   the chracteristic equation and the eigenvalues of this covariance 
   matrix.
 
   Soln:

   The chracteristic equation is:
         -lambda^3 + 14*lambda^2 - 49*lambda + 36
   The eigenvalues are lambda(1)=4 and lambda(2)=9 and lambda(3)=1.