# 程序代写代做代考 KING�S UNIVERSITY COLLEGE

KING�S 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, A�B, 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 = A�1, 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

uk��k

and k~uk =

p

~u � ~u =

vuutk=nX

k=1

uk�uk =

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.

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