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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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