M346 Final Exam Solutions
Given December 13, 2000

Problem 1.

Consider the vector space tex2html_wrap_inline153 of tex2html_wrap_inline155 matrices, let tex2html_wrap_inline157 . Consider the linear transformations tex2html_wrap_inline159 and tex2html_wrap_inline161 .

a) Find the matrix of tex2html_wrap_inline163 relative to the basis


We compute:


so we have


b) Find the matrix of tex2html_wrap_inline169 relative to the same basis.

We compute:


so we have


Problem 2. Let tex2html_wrap_inline173 .

a) Consider the equations tex2html_wrap_inline175 , with A as above. What are the stable and unstable modes? What is the dominant eigenvector?

The eigenvalues are tex2html_wrap_inline179 and tex2html_wrap_inline181 , with eigenvectors tex2html_wrap_inline183 and tex2html_wrap_inline185 . The first mode is stable since tex2html_wrap_inline187 , while the second is unstable since tex2html_wrap_inline189 . The dominant eigenvector is tex2html_wrap_inline181 .

b) Consider the equations tex2html_wrap_inline193 , 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 tex2html_wrap_inline197 is positive or negative. Since tex2html_wrap_inline199 , the first mode is unstable. Since tex2html_wrap_inline201 , the second mode is stable. Now the dominant eigenvector is tex2html_wrap_inline203 .

Problem 3. Let tex2html_wrap_inline205 . Which of the following are Hermitian? Which are unitary? Which are both? Which are neither?

Notice that the eigenvalues of A are tex2html_wrap_inline209 , and the eigenvectors are orthogonal. You can see this by calculating them [they are tex2html_wrap_inline211 and tex2html_wrap_inline213 ], 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) tex2html_wrap_inline223

Hermitian but not unitary, since tex2html_wrap_inline225 are real but not of norm 1.

d) tex2html_wrap_inline227

Unitary but not Hermitian, since tex2html_wrap_inline229 are complex but of norm 1.

Problem 4. In tex2html_wrap_inline231 with the standard inner product, consider the vectors tex2html_wrap_inline233tex2html_wrap_inline235tex2html_wrap_inline237tex2html_wrap_inline239 . Apply Gram-Schmidt to turn this into an orthogonal basis for tex2html_wrap_inline241 .

tex2html_wrap_inline243 .

tex2html_wrap_inline245 .

tex2html_wrap_inline247 .

tex2html_wrap_inline249 .
(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 tex2html_wrap_inline253 .

a) Reduce this 2nd order difference equation to a tex2html_wrap_inline155 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 tex2html_wrap_inline263 and tex2html_wrap_inline265 , so the most general solution is


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

We have tex2html_wrap_inline277 , so tex2html_wrap_inline279 and tex2html_wrap_inline281 . Thus tex2html_wrap_inline283 .

Problem 6. Consider the nonlinear system of equations


a) Linearize this system of equations near the fixed point tex2html_wrap_inline285 .


Defining tex2html_wrap_inline289 , our linearized equations are tex2html_wrap_inline291 .

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

The eigenvalues of A are tex2html_wrap_inline295 , with eigenvectors tex2html_wrap_inline297 . Since both eigenvalues are (in magnitude) bigger than 1, both modes are unstable.

c) Is the fixed point tex2html_wrap_inline285 stable?

And so tex2html_wrap_inline285 is an unstable fixed point.

Problem 7. Diagonalize the matrix tex2html_wrap_inline303

Since it is block triangular, to find the eigenvalues you just need to diagonalize each block. The eigenvalues of the upper left block are tex2html_wrap_inline305 , while the eigenvalues of the lower right block are 4 and -1. The eigenvectors are tex2html_wrap_inline307tex2html_wrap_inline309tex2html_wrap_inline311 ,  b4=(1, 0, 1, -1)^T.  Computing the first two entries of tex2html_wrap_inline315 is messy. I'll accept an answer of
(junk, junk, 3,2 tex2html_wrap_inline317 .

Problem 8.

We wish to solve the differential equation


on the interval tex2html_wrap_inline321 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 tex2html_wrap_inline327 (with Dirichlet boundary conditions).

We already know the eigenvalues and eigenvectors of tex2html_wrap_inline329 , namely tex2html_wrap_inline331 and tex2html_wrap_inline333 . The eigenvalues of tex2html_wrap_inline327 are just one less ( tex2html_wrap_inline337 ) and the eigenvectors are the same.

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

Let tex2html_wrap_inline339 . Then


c) Given the initial conditions tex2html_wrap_inline343tex2html_wrap_inline345 , find f(x,t) for all tex2html_wrap_inline349 and all t.

The only nonzero coefficients are tex2html_wrap_inline353 and tex2html_wrap_inline355 , so