Semester Two Final Examinations, 2019 ENGG7302 Advanced Computational Techniques in Engineering
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This exam paper must not be removed from the venue


School of Information Technology and Electrical Engineering
EXAMINATION
Semester Two Final Examinations, 2019
ENGG7302 Advanced Computational Techniques in Engineering
This paper is for St Lucia Campus students.
Examination Duration: 90 minutes
Reading Time: 10 minutes
Exam Conditions:
This is a Central Examination
This is a Closed Book Examination - no materials permitted
During reading time - write only on the rough paper provided
This examination paper will be released to the Library
Materials Permitted In The Exam Venue:
(No electronic aids are permitted e.g. laptops, phones)
Calculators - Casio FX82 series or UQ approved (labelled)
Materials To Be Supplied To Students:
1 x 14-Page Answer Booklet
Instructions To Students:
Additional exam materials (eg. answer booklets, rough paper) will be
provided upon request.

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Semester Two Final Examinations, 2019 ENGG7302 Advanced Computational Techniques in Engineering
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Part A. (30 marks in total, 5 marks each)
For each question, select the correct answer (only one option is correct among the four
ones; write down your answer in the answer booklets.)
1. Consider the full singular value decomposition (SVD) of a matrix VH, and
A , . Consider the following statements, [1] U, V must be orthogonal
matrices; [2] ; [3] may have min non-zero singular values;
[4] U, V may have the same dimension. Which of the following is correct?
(a) [1], [2], [3], [4]
(b) Only [1], [2], [3]
(c) Only [3],[4]
(d) None of [1], [2], [3], [4]

2. Consider a matrix and its third row vector. What is the difference
between their -norms?
(a) 4
(b) 3
(c) 2
(d) 0

3. Consider the 2-norm and Frobenius norm of a matrix A. Which of the following is
correct?
(a) Its 2-norm is always larger than its Frobenius norm
(b) Its 2-norm is always smaller than its Frobenius norm
(c) Its 2-norm can be larger than its Frobenius norm
(d) Its 2-norm cannot be larger than its Frobenius norm
4. Consider a matrix A . It may have the following properties, [1] A must
have a pseudo-inverse; [2] range(A) ; [3] rank(A) is always equal to m; [4] 0 is
not an eigenvalue of A. Then which of the following is always correct?
(a) Only [2]
(b) Only [1],[3]
(c) [1], [2], [3], [4]
(d) Only [1],[2]

5. If P is a projection matrix, then it may have the following properties, [1] P3=P2; [2]
range[I-P] = null(P), where I is the identity matrix; [3] if P is an orthogonal projector,
then it must be a symmetric matrix; [4] it must be a square matrix. Then which of the
following is always correct?
(a) Only [1],[2],[4]
(b) Only [2],[4]
(c) [1], [2], [3], [4]
(d) Only [3], [4]

6. If A is a unitary matrix, consider the following statements: [1] its singular value
decomposition (SVD) is VH, must be an identity matrix; [2] its eigenvalues
are equal to one. Which of the following is correct?
(a) [1], [2]
(b) Only [1]
(c) Only [2]
(d) Neither [1] nor [2]
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Semester Two Final Examinations, 2019 ENGG7302 Advanced Computational Techniques in Engineering
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Part B. (70 marks in total)

Question 7.
(25 marks)
Consider a 2×2 matrix A . Perform the following calculations.
(a) If
- Calculate its full SVD (that is, find its U, , V );
- Show that U is an unitary matrix;
- Find the null-space, range, rank and condition number of matrix A;
(b) If
Use a low rank approximation (using rank r =1), to form a new matrix B, please
calculate the Frobenius norm of (A-B).

Question 8.
(15 marks)
Find (a) the projection of vector b on the column space of matrix A and
(b) The projection matrix P that projects any vector in (two-element vector) to the
column space of matrix A.
Note:

Question 9.
(20 marks)
Consider a linear system Ax=b, and the SVD of the matrix VH.
- Use matrices U, , V to express the pseudo-inverse of the linear system;
- Show that , where are the first column vectors of matrices U,
V respectively, and is the largest sigular value.

Question 10.
(10 marks)
Consider the following claim:
“There exists a matrix A for which the range and null space are identical.”

If you think such a matrix can exist, give an example and show the matrix’s range and
null space; otherwise, give your reason to reject this claim.

END OF EXAMINATION
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