ENGG7302 Advanced Computational Techniques in Engineering
Semester One 2020 - Final Examination
This is an open book exam – all materials permitted.
Instructions
Answer all the questions.
For further instructions, please refer coversheet.
Part A. (54 marks in total, 6 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 and reduced singular value decompositions (SVD) of a square
matrix = ∑VH, for both SVDs, which of the flowing statements is correct: [1] U, V
must be the same orthogonal matrices; [2] −1 = , = −1; [3] ∑ must be
different from each other; [4] U, V may have the same rank.
(a) [1], [2], [3], [4]
(b) Only [2]
(c) Only [4]
(d) None of [1], [2], [3], [4]
2. Consider the 1-norm, 2-norm, ∞-norm and Frobenius norm of the matrix =
[
1 0 0
0 2 0
0 0 3
], which norm has the largest value?
(a) 1-norm
(b) 2-norm
(c) ∞-norm
(d) Frobenius norm
3. If we use the normal equation method to solve the linear least square (LS) problems
Ax=b, where the null space of A is empty. It may the following properties,
[1] The matrix AHA is invertible; [2] The matrix AHA is singular
[3] The LS solution is (AHA)-1 (AHb). We can definitely say that:
(a) [1], [2], [3] are all correct
(b) Only [1], [2] are correct
(c) Only [1], [3] are correct
(d) [1], [2], [3] are all incorrect
4. Consider the full singular value decompositions (SVD) of a matrix = ∑VH, and if
we do further SVD operation on matrix = ZH, what is the relationship between
the 1-norm of (norm_1()) and the smallest eigenvalues of (eig_s())?
(a) norm_1() > eig_s()
(b) norm_1() = eig_s()
Semester One Final Examination, 2020 ENGG7302 Advanced Computational Techniques in Engineering
Page 2 of 3
(c) norm_1() < eig_s()
(d) none of (a),(b) and (c)
5. For the following matrices: A=
1
1
1
, B=
1 1
1 1
1 1
, C=
1 1 1
1 1 1
1 1 1
, these three matrices may
have the following properties, [1] the same range; [2] the same number of non-zero
singular values; [3] their 2-norm and Frobenius norm are the same; [4] the same
eigenvalue decomposition. Which of the following is correct
(a) [1], [2], [3], [4]
(b) Only [1], [2], [3]
(c) Only [1], [2]
(d) Only [1]
6. Which of the following matrices are unitary matrix (UM) or orthogonal projection
matrix (OPM)?
[1] = [
0
0 1
], [2] = [
0
− 0
] , [3] = [
1 −
1
] , where 2 = −1
(a) [1] is not an OPM and [2] is an UM
(b) none of them are UM
(c) [1] and [3] are UM
(d) [2] and [3] are OPM
7. Suppose a vector v is decomposed into orthogonal components with respect to
orthogonal vectors q1,… qn, so that
1 21 2
0.H H Hn nr v q v q q v q q v q
This implies that
(a) The vectors iq are linearly dependent
(b) is orthogonal to vectors q1,… qn
(c) 0v
(d) 1, nv q q
8. Consider the full SVD of a matrix = ∑VH, it may have the following properties [1]
Only V is unique; [2] U is a unitary matrix; [3] any matrix A’s SVD can be calculated
with the help of eigenvalue decomposition; [4] Only ∑ is unique. Which of the
following is correct
(a) only [2], [4]
(b) [1], [2], [3], [4]
(c) only [2], [3]
(d) only [2], [3], [4]
9. For two vectors u=(0 -1 2)H , v=(1 2 0)H , their inner product and the rank of the outer
product are
(a) -2 and 0
(b) 2 and 0
(c) 0 and 3
(d) -2 and 1
Semester One Final Examination, 2020 ENGG7302 Advanced Computational Techniques in Engineering
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Part B. (46 marks in total)
Question 10 (13 marks)
Consider a real, square ( × ) matrix = T, and its eigenvalues are distinct. Show
that its eigenvectors are orthogonal.
Question 11 (20 marks)
Consider a matrix
= [
2 0
0 0
0 0
]
(a) Compute its full and reduced SVD;
(b) Compute its pseudo-inverse;
(c) Compute its condition number.
Hint: for (a), it is unnecessary to implement detailed SVD calcuations based on
eigenvalue decomposition, you can write down those matrices with essential
explaination.
Question 13 (13 marks)
Given the vectors 1 = [
1
0
1
], 2 = [
3
1
1
], 3 = [
2
−1
3
] , which one best lies in the direction of
= [
1
1
1
]? Explain by calculation.