MATH 2303/3303: Matrix theory and its applications
Course Description and Syllabus
Fall 2014
Instructor: Matt Young
Email: myoung@maths.hku.hk
Oce: Run Run Shaw Building, Room 318
Consultation Hours: 9:30am-11:30am on Tuesdays and and 9:30am-10:30am on Fr

The University of Hong Kong
MATH 2303/3303 Test # 2 Solutions
(Last updated: November 13, 2014)
Name:
UID:
Problem
Points
Scores
1
2
25 10
3
10
4
10
Total:
55
This test has four problems, weighted as shown. Please show your work full credit may
not be giv

The University of Hong Kong
MATH 2303/3303 Test # 1 Solutions
(Last updated: October 10, 2014)
Name:
UID:
Problem
Points
Scores
1
2
21 10
3
10
4
14
Total:
55
This test has four problems, weighted as shown. Please show your work full credit may
not be give

MATH 2303/3303 Partial solutions for practice problems for weeks 10
and 11
These are practice problems for the material discussed in lectures on Wednesday, Nov. 19 and
Wednesday, Nov. 26, 2014. These problems will not be collected.
1. For each of the foll

The University of Hong Kong
MATH 2303/3303 Test # 2
UID:
Name:
Problem
Points
Scores
1
2
25 10
3
10
4
10
Total:
55
This test has four problems, weighted as shown. Please show your work full credit may
not be given if only the answers appear.
1. For each o

The University of Hong Kong
MATH 2303/3303 Test # 1
UID:
Name:
Problem
Points
Scores
1
2
21 10
3
10
4
14
Total:
55
This test has four problems, weighted as shown. Please show your work full credit may
not be given if only the answers appear.
1. [3 marks e

MATH 2303/3303 Practice problems for weeks 10 and 11
These are practice problems for the material discussed in lectures on Wednesday, Nov. 19 and
Wednesday, Nov. 26, 2014. These problems will not be collected.
1. For each of the following matrices A nd a

MATH 2303/3303 Partial solutions for practice problems for week 9
These are practice problems for the material discussed in lecture on Wednesday, Nov. 12, 2014.
These problems will not be collected.
1. For each of the following matrices A nd a Jordan cano

MATH 2303/3303 Partial solutions for practice problems for week 8
These are practice problems for the material discussed in lecture on Wednesday, Nov. 5, 2014.
These problems will not be collected.
1. Use Moore-Penrose inversion to nd the least squares so

MATH 2303/3303 Practice problems for week 9
These are practice problems for the material discussed in lecture on Wednesday, Nov. 12, 2014.
These problems will not be collected.
1. For each of the following matrices A nd a Jordan canonical form J for A. Ma

MATH 2303/3303 Practice problems for week 8
These are practice problems for the material discussed in lecture on Wednesday, Nov. 5, 2014.
These problems will not be collected.
1. Use Moore-Penrose inversion to nd the least squares solutions to each of the

MATH 2303/3303 Practice problems for week 7
These are practice problems for the material discussed in lecture on Wednesday, Oct. 29, 2014.
These problems will not be collected.
1. Find the singular value decomposition and the polar decomposition of each o

MATH 2303/3303 Partial solutions to the practice problems for week 6
These are partial solutions to the practice problems for the material discussed in lecture on
Wednesday, Oct. 22, 2014. They are not meant to be complete solutions.
1. Find the eigenvalu

MATH 2303/3303 Practice problems for week 6
These are practice problems for the material discussed in lecture on Wednesday, Oct. 22, 2014.
These problems will not be collected.
1. Find the eigenvalues of the matrix
A=
a b
b a
M22 (R)
by optimizing the Ra

MATH 2303/3303 Practice problems for week 7
These are practice problems for the material discussed in lecture on Wednesday, Oct. 29, 2014.
These problems will not be collected.
1. Find the singular value decomposition and the polar decomposition of each o

MATH 2303/3303 Practice problems for week 3
These are practice problems for the material discussed in lecture on Wednesday, Sept. 17, 2014.
These problems will not be collected.
1. Let v1 = (1, 0, 1, 0), v2 = (1, 1, 1, 1) and v3 = (0, 2, 4, 2). Apply the

MATH 2303/3303 Practice problems for week 4
These are practice problems for the material discussed in lecture on Wednesday, Sept. 24, 2014.
These problems will not be collected.
1. For each matrix A, nd an orthogonal or unitary matrix Q and a diagonal mat

MATH 2303/3303 Homework 5
This homework will be collected at the end of class on October 29, 2014.
1. Let A Mnn (C) be a Hermitian matrix. Prove the following properties of the Rayleigh
quotient (A, v) =
v Av
:
v 2
(a) (A, cv) = (A, v) if 0 = c C.
(b) If

MATH 2303/3303 Homework 3
This homework will be collected at the end of class on Wednesday, Sept. 24, 2014.
1. Fix positive real numbers 1 , . . . , n . For z = (z1 , . . . , zn ) Cn set
n
z
1
i |zi |,
=
i=1
1
2
n
z
2
i |zi |2
=
i=1
and
z
Which of
1,
2
an

MATH 2303/3303 Homework 8
This is the last homework. It will be collected at the end of class on Nov. 29, 2014.
1. For each of the following matrices A, nd its Jordan canonical form J. For the rst three
parts, nd a matrix X such that A = XJX 1 .
(a)
5 3 2

MATH 2303/3303 Homework 7
This homework will be collected at the end of class on Nov. 19, 2014.
1. For each of the following matrices A, nd a Jordan canonical form J and a matrix X such
that A = XJX 1 .
(a)
A=
2 2
2 2
(b)
2 1 4
A = 0 2 1
0 0 3
(c)
1
0
A=

MATH 2303/3303 Homework 4
This homework will be collected at the end of class on October 8, 2014.
1. For each matrix A, nd an orthogonal or unitary matrix Q and a diagonal matrix D such
that Q AQ = D.
2
1+i
1i
3
0 1 1
(b) A = 1 0 1
1 1 0
2 0 36
(c) A = 0

MATH 2303/3303 Homework 6
This homework will be collected at the end of class on Nov. 5, 2014.
1. Find the singular value decomposition and the polar decomposition of each of the following
matrices:
(a)
A=
2 0
0 0
(b)
1 1
B = 0 1
1 0
(c)
C=
3 2 2
2 3 2
2

MATH 2303/3303 Homework 2
This homework will be collected at the end of class on Wednesday, Sept. 17, 2014.
1. For each complex matrix below, nd its eigenvalues and their geometric and algebraic
multiplicities. Use this to compute the determinant of each

MATH 2303/3303 Homework 1
This homework will be collected at the end of class on Wednesday, Sept. 10, 2014.
1. View C as a vector space over R as described in class. Show that C and R2 are isomorphic
as real vector spaces.
2. Show that Symn (R), the set o