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The University of British Columbia
Final Examination - April 24, 2010
Mathematics 307
Section 201
Closed book examination Time: 2.5 hours
Last Name First Name
Signature Student Number
Rules governing examinations
- Each candidate must be prepared to produce, upon request, a UBC-
card for identification.
- Candidates are not permitted to ask questions of the invigilators,
except in cases of supposed errors or ambiguities in examination ques- tions.
- No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or
to leave during the first half hour of the examination.
- Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and
shall be liable to disciplinary action. (a) Having at the place of writing any books, papers
or memoranda, calculators, computers, sound or image play- ers/recorders/transmitters (including telephones), or other memory
aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candi- dates or imaging devices. The plea of accident or forgetfulness shall
not be received.
- Candidates must not destroy or mutilate any examination material;
must hand in all examination papers; and must not take any exami- nation material from the examination room without permission of the
invigilator.
- Candidates must follow any additional examination rules or direc- tions communicated by the instructor or invigilator.
question score out of
total 100
Special instructions
- Solve 5 out of 6 questions to have a full score. All questions are equal in their weight.
- Indicate which 5 questions are intended for grading by marking the question numbers in the table above. The
draft of the remaining question, if any, will not be graded.
- Make sure to explain any notation introduced in your solution and not given in the formulation of the question.
- Provide complete derivation, as the final answers only are not worth any score.
- No auxiliary material of any kind, including calculators, is allowed.
- Submit the question sheet along with your answers. Make sure that all of your written pages remain intact.
- Verify that your full name, student number and signature appear on the work in ink.
- Below are 6 statements. Mark each one as true or false and give a general explanation ( 3
1 3
points per part ).
If you decide that a statement is false, you can give a contradicting example instead. Bear in mind that to
mark a statement true all of its parts must be true.
(a) It is given that a real matrix A is a root of the equation A
n − A = 0 with n > 1 an integer. Then it is
possible to conclude the following. The size of A is n × n because by the Cayley - Hamilton theorem A
satisfies p(A) = 0, with p(λ) being the characteristic polynomial of A. Since it is given that A is a root of
an n-th degree polynomial, its characteristic polynomial must be of degree n and thus A is n × n.
(b) For any two distinct vectors r 1 , r 2 ∈ R
4 there exists a subspace V ⊂ R
4 such that V ⊥ span{r 1 , r 2 }. In
addition there exists a set of vectors W ⊂ R
4 , such that each vector w ∈ W satisfies w /∈ V and 〈w, ri〉 = 0,
i = { 1 , 2 }. Then the set W is
W = R
4 \ {V ∪ span{r 1 , r 2 }}
and its dimension is
dim W = 1 or dim W = 2.
(c) If the equation Ax = 0 with A ∈ R m×n has a non-trivial solution, then dim Im A T 6 n − 1.
(d) The characteristic polynomial of a matrix A is given by p(λ) = λ 2 − λ. Then using the Cayley - Hamilton
theorem it is possible to compute A − 1 = I.
(e) A projection on a subspace spanned by the columns of a real matrix A is given by P = A(A T A) − 1 A T .
Then if A is an orthogonal matrix, P = AA T .
(f) Pn×n is a projection matrix with n > 2. Then regardless of n it is always possible to construct a quadratic
polynomial such that P is its root.
- A real matrix A 5 × 5 is brought into its Jordan canonical form
B =
by a matrix S 5 × 5 , i.e. A = SBS − 1
. Answer the questions below, substantiate all your statements ( 3
1 3
points
per part ).
(a) What are the eigenvalues of A? Include algebraic multiplicities.
(b) What is the rank of A − I ( I being the 5 × 5 identity matrix )?
(c) What is the dimension of the kernel of A + I ( I being the 5 × 5 identity matrix )?
(d) What is the determinant of the product matrix CA if C ∈ C
5 × 5 is a Hermitian matrix with full geometric
multiplicity?
(e) What is the condition of A?
(f) How should the matrix S be computed? Write an equation for each column, state whether a unique
solution is expected and explain.
- Consider a matrix A ∈ C
n×n , a real diagonal matrix S
2 containing the eigenvalues of A
∗ A ( all positive ) and
a matrix V that diagonalises A
∗ A. Answer the following questions, substantiate all your statements ( 4 points
per part ).
(a) Explain why S
− 1 exists and show that the matrix U = AV S
− 1 is unitary.
(b) Show that U diagonalises AA
∗ with S
2 giving the eigenvalues.
(c) From (a) and (b) it follows that AA
∗ and A
∗ A have the same eigenvalues. How are their eigenvectors
connected?
(d) For the example below
A =
what is ||A||? Explain. Note that here A /∈ C
n×n .
(e) Find a vector on the unit circle
cos θ
sin θ
that is stretched most by A.
- Consider the matrix
P =
α 0 ∗
0 β 0
∗ ∗ γ
(a) ( 5 points ) Fill in the star etnries so that P is a stochastic matrix.
(b) ( 5 points ) Show that the eigenvalues of P are
λ 1 =
, λ 2 = β, λ 3 = α + γ − 1 , λ 4 = 1.
(c) ( 10 points ) Given that β = 1 and α = γ =
, draw the network corresponding to the random walk
represented by P and find the limiting state xn of the walk. Explain the result according to the drawing.
