### 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 R**^{2}
Recall that the general rotation matrix which rotates vectors in the
counterclockwise direction by angle theta is given by

M:=Matrix([[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]);

**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 R^{2},
write out the Eigenvector decomposition of the iterate x_{k}, where the
foxes F_{k}
are the first component of this state vector, and the rabbits
R_{k} the second.

**Part C:**
Use the decomposition to explore what will happen to the
vector x_{k} 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