[SOLVED] KINGS UNIVERSITY COLLEGE

KINGS UNIVERSITY COLLEGE
at the University of Western Ontario

DEPARTMENT OF ECONOMICS, BUSINESS AND
MATHEMATICS

Mathematics 1600b
Forth Quiz, Bonus questions

due date Tuesday, April 4, 2017, 1:30 p.m.

Instructor S.V. Kuzmin

Name (please print)

Student number

Justify your answers by showing su cient work to get the full
marks.
Write your answers that involve complex numbers in a standard form

z = a+ bi, where a and b are real numbers (if you have z1
z2
or z2 etc., convert

it into standard form)
Bonus questions.
Note that the complex conjugate A of a matrix A is the matrix whose

entries are the complex conjugates of the corresponding entries of A.
Let A be a square matrix. A is Hermitian if AT = A, A is skew-

Hermitian if AT = A:(This is generalization of symmetric and skew-
symmetric real matrices which is useful in engineering, mathematics, physics
and chemistry).
The following theorem summarizes the basic properties of such matrices.
Theorem.
1. The main diagonal of a Hermitian matrix consists of real numbers.
2. The main diagonal of a skew-Hermitian matrix consists of zeroes or

pure imaginary numbers.

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3. A matrix that is both Hermitian and skew-Hermitian is a zero matrix.
4. If A and B are Hermitian, so are A + B, A B, and cA for any real

scalar c.
5. If A and B are skew-Hermitian, so are A+B, AB, and cA for any

real scalar c.
B1. (3 marks). Prove this theorem.
Note also that the generalization of orthogonal matrices (the last lecture)

is: a square matrix A is unitary if AT = A1, or equivalently A AT = I:

B2. (2 mark). Determine whether the matrix A =

2
4 1p2 ip2 0ip

2
1p
2
0

0 0 1

3
5 is

unitary matrix and compute the inverse of A:

B2. (5 marks). Prove that the column vectors of a unitary matrix U form
an orthonormal set in Cn (vectors with complex components) with respect

to the complex Euclidean inner product (if ~u; ~ 2 Cn, then ~u ~
k=nX
k=1

ukk

and k~uk =
p
~u ~u =

vuutk=nX
k=1

ukuk =

vuutk=nX
k=1

jukj
2).

B3. (5 marks). Find ALL a; b; and c (a; b; and c 2 C) for which the
matrix

A =
1
p
3

2
4
p
3 0 a
0 1 + i b
0 1 c

3
5

is unitary.

2