Problem Set 6

Problem 1:  7.1   #26 by hand and on Maple via the Eigenvectors(A); command also compare your answers and compare your answers nad resolve any apparent conflicts or differences.
Problem 2:  Rotation matrices in R2   Recall that the general rotation matrix which rotates vectors in the counterclockwise direction by angle theta is given by
  Part A:   Apply the Eigenvalues(M); command. Notice that there are real eigenvalues for certain values of theta only. What are these values of theta and what eigenvalues do they produce? (Recall that I = the square root of negative one does not exist as a real number and that cos(theta) is less than or equal to 1 always.)
  Part B: For each real eigenvalue, find a basis for the corresponding eigenspace.
  Part C:   Use only a geometric explanation to explain why most rotation matrices have no eigenvalues or eigenvectors (ie scaling along the same line through the origin). Address the definition of eigenvalues/eigenvectors in your response as well as how the rotation angle connects to the definition in this case.
Problem 3:  Foxes and Rabbits (Predator-prey model)
Suppose a system of foxes and rabbits is given as:

  Part A: First compute the eigenvectors in Maple. Then show that the eigenvectors satisfy the definitions of span and li.
  Part B: Since they form a basis for R2, write out the Eigenvector decomposition of the iterate xk, where the foxes Fk are the first component of this state vector, and the rabbits Rk the second.
  Part C: Use the decomposition to explore what will happen to the vector xk in the longterm, and what kind of vector(s) it will travel along to achieve that longterm behavior, and then fill in the blanks:
If ___ equals 0 then we die off along the line____ [corresponding to the eigenvector____], and otherwise we [choose one: die off or grow or hit and then stayed fixed] along the line____ [corresponding to the the eigenvector____].
  Part D: Determine a value to replace 1.05 in the original system that leads to constant levels of the fox and rabbit populations (ie an eigenvalue of 1), so that eventually neither population is changing. What is the ratio of the sizes of the populations in this case?
Problem 4:  7.2   7
Problem 5:  7.2 14
Problem 6:  7.2 20