MTH208 Advanced Linear Algebra

MTH208 Advanced Linear Algebra

MTH208
Advanced Linear Algebra
Tutor-Marked Assignment
July 2021 Presentation
MTH208 Tutor-Marked Assignment
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 2 of 6
TUTOR-MARKED ASSIGNMENT (TMA)
This assignment is worth 20% of the final mark for MTH208 Advanced Linear Algebra.
The cut-off date for this assignment is 25 October 2021 (Monday), 23 55 hours.
Note to Students:

MTH208 Advanced Linear Algebra

You are to include the following particulars in your submission: Course Code, Title of the
TMA, SUSS PI No., Your Name, and Submission Date.
For example, ABC123_TMA01_Sally001_TanMeiMeiSally (omit D/O, S/O). Use underscore
and not space.
Question 1
(a) Calculate the Jordan Canonical Form of the matrix
(

MTH208 Advanced Linear Algebra

−3 0 0 0 0 0
0 −3 0 0 0 0
0 0 −3 0 0 0
0 0 0 2 0 0
0 −1 1 1 2 0
0 −1 −1 1 −1 4)

.
(14 marks)
(b) Write down 2 possible Jordan Caonical Forms for an 8 × 8 matrix with characteristic
polynomial (𝑥 + 1)
2
(𝑥 − 3)
6
and minimum polynomial (𝑥 + 1)(𝑥 − 3)
2
.

MTH208 Advanced Linear Algebra

(6 marks)
MTH208 Tutor-Marked Assignment
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 3 of 6
Question 2
Let
𝐴 = (
−82/75 26/75
−58/15 44/15
−151/75 368/75)
and let 𝐵 = 𝐴
∗𝐴, a positive definite matrix.
(a) Illustrate the property of normal operators by providing a unitary matrix 𝑄 and a
diagonal matrix 𝐷 such that 𝑄
∗𝐵𝑄 = 𝐷.
(5 marks)
(b) Find a singular value decomposition of 𝐴.

MTH208 Advanced Linear Algebra

(5 marks)
(c) Determine 2 × 2 matrices 𝐶1, 𝐶2 such that 𝐶1 is Hermitian, 𝐶2 is not Hermitian, and
(𝐶1)
∗𝐶1 = 𝐵 = (𝐶2)
∗𝐶2. Justify your answer fully.
(6 marks)
(d) Let 𝐺 be a complex 𝑚 × 𝑛 matrix with singular value decomposition 𝐺 = 𝑈Σ𝑉
∗
, where
𝑚 > 𝑛.
Let
𝐻 = (
0 𝐺
∗
𝐺 0
)
be a block matrix of size (𝑚 + 𝑛) × (𝑚 + 𝑛), where the 0’s represent zero matrices.
(i) Give a brief explanation why all the eigenvalues of 𝐻 are real, and at least one
of its eigenvalues is 0.
(2 marks)
(ii) Find a unitary matrix 𝑃 and a diagonal matrix 𝐸 such that 𝑃
∗𝐻𝑃 = 𝐸. Express
𝑃 and 𝐸 in terms of the matrices 𝑈,𝑉 and Σ.
(7 marks)
MTH208 Tutor-Marked Assignment
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 4 of 6
Question 3
(a) Let
𝑤1 = (
13
8
−3
8
) , 𝑤2 = (
−11
4
1
−6
) , 𝑤3 = (
2
3
−4
1
) , 𝑣1 = (
3
4
−1
2
) , 𝑣2 = (
−2
2
0
−1
)

MTH208 Advanced Linear Algebra

and let 𝑊 = span{𝑤1, 𝑤2, 𝑤3}. We equip ℝ4 with the standard inner product.
(i) Show that 𝑣1 and 𝑣2 are orthogonal and that they are elements of 𝑊.
(5 marks)
(ii) Find a vector 𝑣3 such that {𝑣1, 𝑣2, 𝑣3} is an orthogonal basis of 𝑊.
(4 marks)
(b) Suppose that 𝑥1, 𝑥2, 𝑥3, 𝑥4 are linearly independent vectors in ℝ5
such that
〈𝑥1, 𝑥1
〉 = 9,〈𝑥1, 𝑥2
〉 = 9,〈𝑥1, 𝑥3
〉 = 1,〈𝑥1, 𝑥4
〉 = −4,
〈𝑥2, 𝑥2
〉 = 36,〈𝑥2, 𝑥3
〉 = −3,〈𝑥2, 𝑥4
〉 = −3,
〈𝑥3, 𝑥3
〉 = 16,〈𝑥3, 𝑥4
〉 = 9,〈𝑥4, 𝑥4
〉 = 25,
and let 𝑈 = span{𝑥1, 𝑥2, 𝑥3, 𝑥4}. We equip ℝ5 with the standard inner product.
(i) Use the Gram-Schmidt process to find vectors 𝑦1, 𝑦2, 𝑦3, 𝑦4 such that
{𝑦1, 𝑦2, 𝑦3, 𝑦4
} is an orthogonal basis of 𝑈, expressing each of the vectors
𝑦1, 𝑦2, 𝑦3, 𝑦4 as linear combinations of 𝑥1, 𝑥2, 𝑥3, 𝑥4.
(8 marks)
(ii) Let 𝑇:𝑈 ⟶ 𝑈 be a linear operator such that
𝑇(𝑥1
) = −𝑥1 + 2𝑥2 + 5𝑥3 + 𝑥4
𝑇(𝑥2
) = 2𝑥1 − 𝑥2 + 𝑥3 + 𝑥4
𝑇(𝑥3
) = 3𝑥1 + 4𝑥2 + 2𝑥3 − 𝑥4
𝑇(𝑥4
) = 3𝑥1 + 2𝑥2 + 𝑥3
It is known that 𝑇 is invertible. Let ℬ = {𝑥1, 𝑥2, 𝑥3, 𝑥4} and let 𝒞 =
{𝑦1, 𝑦2, 𝑦3, 𝑦4
} be the bases of 𝑈 as determined in Question 3(b)(i). Compute the
matrix representations of 𝑇
−1
relative to the bases ℬ and 𝒞 respectively.
(8 marks)
MTH208 Tutor-Marked Assignment
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 5 of 6
Question 4
Let 𝑉 denote the real vector space of polynomials of degree at most 3 over ℝ. Define the
polynomials
𝑝1
(𝑥) = 1 + 3𝑥 − 4𝑥
2 + 3𝑥
3
𝑝2
(𝑥) = 2 + 5𝑥 + 2𝑥
2 − 2𝑥
3
𝑝3
(𝑥) = −2 − 11𝑥
2 + 𝑥
3
and let 𝑊 denote the subspace of 𝑉 spanned by 𝑝1
(𝑥), 𝑝2
(𝑥), 𝑝3
(𝑥).
(a) State the dimension of 𝑉 and show that ℬ = {𝑝1
(𝑥), 𝑝2
(𝑥), 𝑝3
(𝑥)} is a basis of 𝑊.
(4 marks)
(b) Find the dual basis ℬ
∗ of ℬ.
(3 marks)
(c) Find a basis of the annihilator 𝑊0 of 𝑊.
(3 marks)
Question 5
Let 𝐴 be a complex 𝑚 × 𝑛 matrix with rank 𝑛, where 𝑚 > 𝑛.
Define
𝐻 = 𝐴(𝐴
∗𝐴)
−1𝐴
∗
.
We equip ℂ
𝑚 with the standard inner product.
(a) Show that 𝐻 and 𝐼 − 𝐻 are self-adjoint matrices that are idempotent, where 𝐼 represents
the 𝑚 × 𝑚 identity matrix.
(4 marks)
(b) Let 𝑇 ∶ ℂ
𝑚 ⟶ ℂ
𝑚 be the orthogonal projection operator of ℂ
𝑚 onto the column space
of 𝐴. Prove that the matrix representation of 𝑇 relative to the standard basis is given by
𝐻.
(3 marks)
(c) Let the (𝑖,𝑗)-entry of 𝐴 be denoted by 𝑎𝑖𝑗 and let 𝑣𝑗 denote the 𝑗
th column of 𝐴. If the
columns of 𝐴 are mutually orthogonal, show that the (𝑖,𝑗)-entry of 𝐻 is given by
∑
𝑎𝑖𝑝𝑎𝑗𝑝
‖𝑣𝑝‖
2
𝑛
𝑝=1
.
(5 marks)
MTH208 Tutor-Marked Assignment
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 6 of 6
Question 6
Let 𝑇 ∶ 𝑉 ⟶ 𝑉 be a linear operator on a finite dimensional complex inner product space 𝑉.
Suppose that 𝑇 is a normal operator. If there is some complex scalar 𝜆, some unit vector 𝑣, and
some positive real number 𝜖 such that
‖𝑇(𝑣) − 𝜆𝑣‖ < 𝜖 ,
show that there is some eigenvalue 𝜇 of 𝑇 such that
|𝜇 − 𝜆| < 𝜖 .
(8 marks)
—- END OF ASSIGNMENT —-

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