Eigenvector hw

1. Review the Healthy Sick worker problem from Problem Set 3 (solutions are on ASULearn, for example).
   
   
2. Summarize in your own words how we calculated the steady state vector on that problem set.
   
   
3. Read through the following definition:
   Definitions If a square matrix N, a column vector x, and a real number lambda satisfy the equation N x=lambda x [which is equivalent to (N- lambda I) x = 0], then x is an eigenvector and lambda is an eigenvalue.
   
   
4. Rewrite the definition when lambda = 1, ie substitute 1 for every place you see lambda, and write this out.
   
   
5. Apply the language of eigenvectors to the computational work we have already completed in problem set 3. In this problem, lambda = 1, and we have already solved for (N-I)x=0 [ie Nx=1x]. So write down the eigenvector here, which is the steady state vector you found on the problem set.
   
   
6. In general, using material from 4.1, how does scalar multiplication
   lambda x
   relate to
   x
   geometrically and algebraically [again x is a column vector, say in the plane, and lambda is a real number].
   
   
7. Read through the Final Research Sessions for Dec 12th, and write down any questions you have.