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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