Problem 2:

1 mark: getting the eigenvalues
            1,1,7

2 marks: getting the eigenvectors
         e.g. (-1 0 1) (-1 1 0) (1 1 1)

3 marks: getting an orthogonal set of eigenvectors
   (1 1 -2)  (-1 1 0)  (1 1 1)
            

2 marks: normalizing the eigenvectors

1 mark: writing down the matrix P

1 mark: writing down D and it agrees with P (i.e. correct order of the
diagonal entries)
 e.g.

    0.4082    0.7071    0.5774
    0.4082   -0.7071    0.5774
   -0.8165         0    0.5774


d =

    1.0000         0         0
         0    1.0000         0
         0         0    7.0000

[Marks for calculation errors are deducted from the total score:
typically one or two marks are deducted if there are some calculation
errors.]

------------Comments:

Generally the marks for Q #2 were fairly good.  The
majority of the students knew how to do this question.  Students did
well on this one since it was so "calculation" oriented, and it was a familiar
question to them from lectures/assignments etc.  A lot of students who
seemed to be doing poorly on the rest of the exam could still do this
question.

A fairly common error was to just diagonalize the matrix instead of getting
an orthogonal diagonalization.  (worth 5/10)

Next most common was forgetting to normalize the vectors (worth 8/10) or
forgetting to get orthogonal eigenvectors (7/10).
Surprising: many students remembered to do one of these
things (i.e. get an orthogonal set of eigenvectors, or normalize), but not
the other.  There was *not* a good grasp of what an orthogonal matrix is.

Some people failed to get the eigenvalues correctly.  For some reason 
1,3,5 was a common answer for the eigenvalues.  Several went along and
calculated [0,0,0] for an eigenvector and didn't notice anything wrong with
this!

There were some who didn't get the eigenvalues (had problems with the
calculating and/or factoring the char polynomial) and went on to describe in
words what they would have done to solve the problem.  I thought about this
for a while and decided that this explanation shouldn't be worth any marks.
The reason I decided this was that I didn't want to take away from the
students who actually went through the calculation correctly.

Some had difficulty with the notion of linearly independent eigenvectors and
found either 1 eigenvectors or 3 eigenvectors to go with the eigenvalue 1
(there should have been 2 lin. indep. ones)

Some forgot to write down the matrix D, or put the eigenvalues in the wrong
order (compared to P).  A few wrote D = diag(1,7) which is only a 2x2
matrix.

The most costly error was: students who actually
calculated P^(-1) instead of realizing that you could write down P^T instead
or (more importantly!) realizing that you never even need to calculate
P^(-1) at all!  "Costly" because it wastes a lot of time that could
have been used on other questions.

The most common *computational* error was incorrectly computing
eigenvectors.  Quite a few couldn't get eigenvectors from a (correctly) row
reduced co-efficient matrix (A - lambda * I).  Next most common were failing
to compute or factor the char poly correctly, and making a mistake when
using Gram-Schmidt.

This question was generally well done and
as a (very rough) estimate: over half of the students
got 7/10 or higher.