M346 Final Exam Solutions
Given December 13, 2000

Problem 1.

Consider the vector space  of  matrices, let  . Consider the linear transformations  and  .

a) Find the matrix of  relative to the basis

We compute:

so we have

b) Find the matrix of  relative to the same basis.

We compute:

so we have

Problem 2. Let  .

a) Consider the equations  , with A as above. What are the stable and unstable modes? What is the dominant eigenvector?

The eigenvalues are  and  , with eigenvectors  and  . The first mode is stable since  , while the second is unstable since  . The dominant eigenvector is  .

b) Consider the equations  , with A as above. What are the stable and unstable modes? What is the dominant eigenvector?

Now the question is whether the real part of  is positive or negative. Since  , the first mode is unstable. Since  , the second mode is stable. Now the dominant eigenvector is  .

Problem 3. Let  . Which of the following are Hermitian? Which are unitary? Which are both? Which are neither?

Notice that the eigenvalues of A are  , and the eigenvectors are orthogonal. You can see this by calculating them [they are  and  ], or from the fact that A is manifestly Hermitian. The various operations all give matrices with the same eigenvectors as A, but different eigenvalues. Since the eigenvectors are orthogonal, a matrix will be Hermitian if its eigenvalues are real, and unitary if its eigenvalues have norm one.

a) A

Both Hermitian and unitary, since 1 and -1 are both real and of norm 1.

b) A + I

Hermitian but not unitary, since 2 and 0 are real but not of norm 1.

c)

Hermitian but not unitary, since  are real but not of norm 1.

d)

Unitary but not Hermitian, since  are complex but of norm 1.

Problem 4. In  with the standard inner product, consider the vectors  . Apply Gram-Schmidt to turn this into an orthogonal basis for  .

.

.

.

.
(sorry...that got cut off by the latex2html program.  The final answer is y4=(0,-1,1,0)^T.

Problem 5. Consider a sequence of numbers satisfying the second order difference equation x(n) = 2 x(n-1) + 3 x(n-2) for  .

a) Reduce this 2nd order difference equation to a  system of first order difference equations.

Let y(n)=x(n-1). Then x(n) = 2x(n-1) + 3y(n-1), and we have

b) Find the most general solution to the first order system.

The eigenvalues of the matrix are 3 and -1, with eigenvectors  and  , so the most general solution is

c) From initial data x(0)=2, x(1)=2, find x(n) for all n.

We have  , so  and  . Thus  .

Problem 6. Consider the nonlinear system of equations

a) Linearize this system of equations near the fixed point  .

Defining  , our linearized equations are  .

b) Find the modes and determine which are stable and which are unstable.

The eigenvalues of A are  , with eigenvectors  . Since both eigenvalues are (in magnitude) bigger than 1, both modes are unstable.

c) Is the fixed point  stable?

And so  is an unstable fixed point.

Problem 7. Diagonalize the matrix

Since it is block triangular, to find the eigenvalues you just need to diagonalize each block. The eigenvalues of the upper left block are  , while the eigenvalues of the lower right block are 4 and -1. The eigenvectors are  ,  b4=(1, 0, 1, -1)^T.  Computing the first two entries of  is messy. I'll accept an answer of
(junk, junk, 3,2  .

Problem 8.

We wish to solve the differential equation

on the interval  with Dirichlet boundary conditions:

for all t. [This is called the Klein-Gordon equation, and comes up in relativistic quantum mechanics. We have not studied this equation, but you can solve it using the same ideas that gave us standing waves solutions to the wave equation.]

a) Find the eigenvalues and eigenfunctions of the operator  (with Dirichlet boundary conditions).

We already know the eigenvalues and eigenvectors of  , namely  and  . The eigenvalues of  are just one less (  ) and the eigenvectors are the same.

b) Find the most general solution to (KG).

Let  . Then

c) Given the initial conditions  , find f(x,t) for all  and all t.

The only nonzero coefficients are  and  , so