Download 8.1 Complex Numbers

9781133110873_0801.qxp
3/10/12
8
8.1
8.2
8.3
8.4
8.5
6:52 AM
Page 391
Complex Vector
Spaces
Complex Numbers
Conjugates and Division of Complex Numbers
Polar Form and DeMoivre’s Theorem
Complex Vector Spaces and Inner Products
Unitary and Hermitian Matrices
Quantum Mechanics (p. 425)
Signal Processing (p. 417)
Elliptic Curve Cryptography (p. 406)
Mandelbrot Set (p. 400)
Electric Circuits (p. 395)
391
Clockwise from top left, James Thew/Shutterstock.com; Mikhail/Shutterstock.com;
Sybille Yates/Shutterstock.com; Lisa F. Young/Shutterstock; Štepán Kápl/Shutterstock.com
9781133110873_0801.qxp
392
3/10/12
Chapter 8
6:52 AM
Page 392
Complex Vector Spaces
8.1 Complex Numbers
Use the imaginary unit i to write complex numbers.
Graphically represent complex numbers in the complex plane as
points and as vectors.
Add and subtract two complex numbers, and multiply a complex
number by a real scalar.
Multiply two complex numbers, and use the Quadratic Formula to
find all zeros of a quadratic polynomial.
Perform operations with complex matrices, and find the determinant
of a complex matrix.
COMPLEX NUMBERS
REMARK
When working with products
involving square roots of
negative numbers, be sure to
convert to a multiple of i before
multiplying. For instance,
consider the following
operations.
冪1冪1 i
i
i2
1 Correct
冪1冪1 冪共1兲共1兲
冪1
1
Incorrect
So far in the text, the scalar quantities used have been real numbers. In this chapter, you
will expand the set of scalars to include complex numbers.
In algebra it is often necessary to solve quadratic equations such as
x 2 3x 2 0. The general quadratic equation is ax 2 bx c 0, and its
solutions are given by the Quadratic Formula
x
b 冪b2 4ac
2a
where the quantity under the radical, b2 4ac, is called the discriminant. If
b2 4ac 0, then the solutions are ordinary real numbers. But what can you conclude
about the solutions of a quadratic equation whose discriminant is negative? For
instance, the equation x2 4 0 has a discriminant of b2 4ac 16, but there is
no real number whose square is 16. To overcome this deficiency, mathematicians
invented the imaginary unit i, defined as
i 冪1
where i 2 1. In terms of this imaginary unit, 冪16 4冪1 4i.
With this single addition of the imaginary unit i to the real number system, the
system of complex numbers can be developed.
Definition of a Complex Number
If a and b are real numbers, then the number
a bi
is a complex number, where a is the real part and bi is the imaginary part of
the number. The form a bi is the standard form of a complex number.
Some examples of complex numbers written in standard form are 2 2 0i,
4 3i, and 6i 0 6i. The set of real numbers is a subset of the set of complex
numbers. To see this, note that every real number a can be written as a complex
number using b 0. That is, for every real number, a a 0i.
A complex number is uniquely determined by its real and imaginary parts. So, two
complex numbers are equal if and only if their real and imaginary parts are equal. That
is, if a bi and c di are two complex numbers written in standard form, then
a bi c di
if and only if a c and b d.
9781133110873_0801.qxp
3/10/12
6:52 AM
Page 393
8.1
Imaginary
axis
Complex Numbers
393
THE COMPLEX PLANE
(a, b) or a + bi
b
a
Real
axis
Because a complex number is uniquely determined by its real and imaginary parts,
it is natural to associate the number a bi with the ordered pair 共a, b兲. With this
association, complex numbers can be represented graphically as points in a coordinate
plane called the complex plane. This plane is an adaptation of the rectangular
coordinate plane. Specifically, the horizontal axis is the real axis and the vertical axis
is the imaginary axis. The point that corresponds to the complex number a bi is
共a, b兲, as shown in Figure 8.1.
The Complex Plane
Plotting Numbers in the Complex Plane
Figure 8.1
Plot each number in the complex plane.
a. 4 3i
b. 2 i
c. 3i
d. 5
SOLUTION
Figure 8.2 shows the numbers plotted in the complex plane.
a.
Imaginary
axis
Imaginary
axis
b.
2
4
4 + 3i
or (4, 3)
3
2
1
−3 −2 −1
1
−2 −1
1
2
3
4
2
3
Real
axis
−2 − i
or (−2, −1)
Real
axis
−3
−2
−4
Imaginary
axis
c.
1
d.
Imaginary
axis
2
4
1
−3 −2 −1
1
2
3
2
1
−2
−3
3
Real
axis
−3i or (0, −3)
−4
−1
5 or (5, 0)
1
2
3
4
5
Real
axis
−2
Figure 8.2
Another way to represent the complex number a bi is as a vector whose
horizontal component is a and whose vertical component is b. (See Figure 8.3.) (Note
that the use of the letter i to represent the imaginary unit is unrelated to the use of i to
represent a unit vector.)
Imaginary
axis
1
Horizontal
component
Real
axis
−1
−2
Vertical
component
4 − 2i
−3
Vector Representation of a Complex Number
Figure 8.3
9781133110873_0801.qxp
394
3/10/12
Chapter 8
6:52 AM
Page 394
Complex Vector Spaces
ADDITION, SUBTRACTION, AND SCALAR
MULTIPLICATION OF COMPLEX NUMBERS
Because a complex number consists of a real part added to a multiple of i, the
operations of addition and multiplication are defined in a manner consistent with the
rules for operating with real numbers. For instance, to add (or subtract) two complex
numbers, add (or subtract) the real and imaginary parts separately.
Definition of Addition and Subtraction of Complex Numbers
The sum and difference of
a bi
and c di
are defined as follows.
REMARK
共a bi兲 共c di兲 共a c兲 共b d 兲i
Sum
共a bi兲 共c di兲 共a c兲 共b d 兲i
Difference
Note in part (a) of Example 2
that the sum of two complex
numbers can be a real number.
Adding and Subtracting Complex Numbers
a. 共2 4i兲 共3 4i兲 共2 3兲 共4 4兲i
5
b. 共1 3i兲 共3 i兲 共1 3兲 共3 1兲i
2 4i
Using the vector representation of complex numbers, you can add or subtract two
complex numbers geometrically using the parallelogram rule for vector addition, as
shown in Figure 8.4.
Imaginary
axis
Imaginary
axis
z = 3 + 4i
4
2
3
1
−1
w=3+i
1
2
z+w=5
1
2
3
4
5
6
−2
Real
axis
−3
1
−3
−3
−4
w = 2 − 4i
Addition of Complex Numbers
2
3
Real
axis
z = 1 − 3i
z − w = −2 − 4i
Subtraction of Complex Numbers
Figure 8.4
Many of the properties of addition of real numbers are valid for complex numbers
as well. For instance, addition of complex numbers is both associative and commutative.
Moreover, to find the sum of three or more complex numbers, extend the definition of
addition in the natural way. For example,
共2 i兲 共3 2i兲 共2 4i兲 共2 3 2兲 共1 2 4兲i
3 3i.
9781133110873_0801.qxp
3/10/12
6:52 AM
Page 395
8.1
Complex Numbers
395
Another property of real numbers that is valid for complex numbers is the
distributive property of scalar multiplication over addition. To multiply a complex
number by a real scalar, use the definition below.
Definition of Scalar Multiplication
If c is a real number and a bi is a complex number, then the scalar multiple
of c and a bi is defined as
c共a bi兲 ca cbi.
Scalar Multiplication with Complex Numbers
a. 3共2 7i兲 4共8 i兲 6 21i 32 4i
38 17i
b. 4共1 i兲 2共3 i兲 3共1 4i兲 4 4i 6 2i 3 12i
1 6i
Geometrically, multiplication of a complex number by a real scalar corresponds to
the multiplication of a vector by a scalar, as shown in Figure 8.5.
Imaginary
axis
Imaginary
axis
3
4
2
3
2z = 6 + 2i
2
1
−1
z=3+i
1
z=3+i
1
2
3
4
5
6
Real
axis
1
−z = −3 − i
2
3
Real
axis
−2
−3
−2
Multiplication of a Complex Number by a Real Number
Figure 8.5
With addition and scalar multiplication, the set of complex numbers forms a
vector space of dimension 2 (where the scalars are the real numbers). You are asked to
verify this in Exercise 55.
LINEAR
ALGEBRA
APPLIED
Complex numbers have some useful applications in
electronics. The state of a circuit element is described by
two quantities: the voltage V across it and the current I
flowing through it. To simplify computations, the circuit
element’s state can be described by a single complex
number z V li, of which the voltage and current are
simply the real and imaginary parts. A similar notation can
be used to express the circuit element’s capacitance and
inductance.
When certain elements of a circuit are changing with
time, electrical engineers often have to solve differential
equations. These can often be simpler to solve using
complex numbers because the equations are less
complicated.
Adrio Communications Ltd/shutterstock.com
9781133110873_0801.qxp
396
3/10/12
Chapter 8
6:52 AM
Page 396
Complex Vector Spaces
MULTIPLICATION OF COMPLEX NUMBERS
The operations of addition, subtraction, and scalar multiplication of complex numbers
have exact counterparts with the corresponding vector operations. By contrast, there is
no direct vector counterpart for the multiplication of two complex numbers.
Definition of Multiplication of Complex Numbers
The product of the complex numbers a bi and c di is defined as
共a bi兲共c di兲 共ac bd 兲 共ad bc兲i.
Rather than try to memorize this definition of the product of two complex numbers,
simply apply the distributive property, as follows.
TECHNOLOGY
Many graphing utilities and
software programs can
calculate with complex
numbers. For example, on
some graphing utilities, you
can express a complex number
a bi as an ordered pair 共a, b兲.
Try verifying the result of
Example 4(b) by multiplying
共2, 1兲 and 共4, 3兲. You should
obtain the ordered pair 共11, 2兲.
共a bi兲共c di兲 a共c di兲 bi共c di兲
Distributive property
ac 共ad 兲i 共bc兲i 共bd 兲i
Distributive property
ac 共ad 兲i 共bc兲i 共bd 兲共1兲
Use i 2 1.
ac bd 共ad 兲i 共bc兲i
Commutative property
共ac bd 兲 共ad bc兲i
Distributive property
2
Multiplying Complex Numbers
a. 共2兲共1 3i兲 2 6i
b. 共2 i兲共4 3i兲 8 6i 4i 3i 2
8 6i 4i 3共1兲
8 3 6i 4i
11 2i
Complex Zeros of a Polynomial
Use the Quadratic Formula to find the zeros of the polynomial p共x兲 x 2 6x 13
and verify that p共x兲 0 for each zero.
SOLUTION
Using the Quadratic Formula,
x
b 冪b2 4ac
6 冪16 6 4i
3 2i.
2a
2
2
Substitute each value of x into the polynomial p共x兲 to verify that p共x兲 0.
REMARK
A well-known result from
algebra states that the complex
zeros of a polynomial with real
coefficients must occur in
conjugate pairs. (See Review
Exercise 81.)
p共3 2i兲 共3 2i兲2 6共3 2i兲 13
共3 2i兲共3 2i兲 6共3 2i兲 13
9 6i 6i 4 18 12i 13 0
p共3 2i兲 共3 2i兲2 6共3 2i兲 13
共3 2i兲共3 2i兲 6共3 2i兲 13
9 6i 6i 4 18 12i 13 0
In Example 5, the two complex numbers 3 2i and 3 2i are complex
conjugates of each other (together they form a conjugate pair). More will be said
about complex conjugates in Section 8.2.
9781133110873_0801.qxp
3/10/12
6:52 AM
Page 397
8.1
Complex Numbers
397
COMPLEX MATRICES
Now that you are able to add, subtract, and multiply complex numbers, you can apply
these operations to matrices whose entries are complex numbers. Such a matrix is
called complex.
Definition of a Complex Matrix
A matrix whose entries are complex numbers is called a complex matrix.
All of the ordinary operations with matrices also work with complex matrices, as
demonstrated in the next two examples.
Operations with Complex Matrices
Let A and B be the complex matrices
A
冤2 3ii
1i
4
冥
and
B
冤2ii
冥
0
1 2i
and determine each of the following.
b. 共2 i兲B
a. 3A
c. A B
d. BA
SOLUTION
1i
3i 3 3i
4
6 9i
12
冤2 3ii
冥 冤
a. 3A 3
0
2 4i
0
1 2i
1 2i 4 3i
冤2ii
冥 冤
b. 共2 i兲B 共2 i兲
c. A B 冤2 3ii
冥
冥
1i
2i
0
3i 1 i
4
i 1 2i
2 2i 5 2i
冥 冤
冥 冤
冥
i 1i
冤 i 1 2i冥冤2 3i 4冥
2 0
2i 2 0
冤
1 2 3i 4i 6
i 1 4 8i冥
2 2 2i
冤
7i
3 9i冥
d. BA 2i
0
Finding the Determinant of a Complex Matrix
Find the determinant of the matrix
TECHNOLOGY
Many graphing utilities and
software programs can perform
matrix operations on complex
matrices. Try verifying the
calculation of the determinant
of the matrix in Example 7.
You should obtain the same
answer, 共8, 26兲.
A
冤2 4i3
SOLUTION
ⱍ
冥
2
.
5 3i
ⱍ
2 4i
2
3 5 3i
共2 4i兲共5 3i兲 共2兲共3兲
10 20i 6i 12 6
8 26i
det共A兲 9781133110873_0801.qxp
398
3/10/12
Chapter 8
6:52 AM
Page 398
Complex Vector Spaces
8.1 Exercises
In Exercises 1–6, determine
the value of the expression.
1. 冪2冪3
2. 冪8冪8
3. 冪4冪4
3
4
4. i
5. i
6. 共i兲7
Simplifying an Expression
In Exercises 7–10,
determine x such that the complex numbers in each pair
are equal.
7. x 3i, 6 3i
8. 共2x 8兲 共x 1兲i, 2 4i
9. 共x 2 6兲 共2x兲i, 15 6i
10. 共x 4兲 共x 1兲i, x 3i
Equality of Complex Numbers
In Exercises 11–16, plot
the number in the complex plane.
11. z 6 2i
12. z 3i
13. z 5 5i
14. z 7
15. z 1 5i
16. z 1 5i
Plotting Complex Numbers
Adding and Subtracting Complex Numbers In
Exercises 17–24, find the sum or difference of the
complex numbers. Use vectors to illustrate your answer.
17. 共2 6i兲 共3 3i兲
18. 共1 冪2i兲 共2 冪2i兲
19. 共5 i兲 共5 i兲
20. i 共3 i兲
21. 6 共2i兲
22. 共12 7i兲 共3 4i兲
23. 共2 i兲 共2 i兲
24. 共2 i兲 共2 i兲
Scalar Multiplication In Exercises 25 and 26, use
vectors to illustrate the operations geometrically. Be sure
to graph the original vector.
25. u and 2u, where u 3 i
26. 3u and 32u, where u 2 i
Multiplying Complex Numbers
find the product.
27. 共5 5i兲共1 3i兲
29. 共冪7 i兲共冪7 i兲
31. 共a bi兲2
33. 共1 i兲3
28.
30.
32.
34.
In Exercises 27–34,
共3 i兲共 23 i兲
共4 冪2i兲共4 冪2 i兲
共a bi兲共a bi兲
共2 i兲共2 2i兲共4 i兲
In Exercises 35–40, determine all the
zeros of the polynomial function.
35. p共x兲 2x 2 2x 5 36. p共x兲 x 2 x 1
37. p共x兲 x 2 5x 6
38. p共x兲 x 2 4x 5
39. p共x兲 x 4 16
40. p共x兲 x 4 10x 2 9
Finding Zeros
Finding Zeros In Exercises 41–44, use the given zero
to find all zeros of the polynomial function.
41. p共x兲 x3 3x2 4x 2
Zero: x 1
42. p共x兲 x3 2x2 11x 52
43. p共x兲 2x3 3x2 50x 75
44. p共x兲 x3 x2 9x 9
Zero: x 4
Zero: x 5i
Zero: x 3i
Operations with Complex Matrices In Exercises
45–54, perform the indicated matrix operation using the
complex matrices A and B.
Aⴝ
45.
47.
49.
51.
53.
冤21ⴚⴙ2ii
冥
1
ⴚ3i
AB
2A
2iA
det共A B兲
5AB
and
Bⴝ
46.
48.
50.
52.
54.
冤1 ⴚⴚ3i
冥
3i
ⴚi
BA
1
2B
1
4 iB
det共B兲
BA
55. Proof Prove that the set of complex numbers, with the
operations of addition and scalar multiplication (with
real scalars), is a vector space of dimension 2.
56.
Consider the functions
p共x兲 x 2 6x 10 and q共x兲 x 2 6x 10.
(a) Without graphing either function, determine
whether the graphs of p and q have x-intercepts.
Explain your reasoning.
(b) For which of the given functions is x 3 i a
zero? Without using the Quadratic Formula, find the
other zero of this function and verify your answer.
57. (a) Evaluate i n for n 1, 2, 3, 4, and 5.
(b) Calculate i 2010.
(c) Find a general formula for i n for any positive integer n.
58. Let A 冤0i 0i 冥.
(a) Calculate An for n 1, 2, 3, 4, and 5.
(b) Calculate A2010.
(c) Find a general formula for An for any positive
integer n.
True or False? In Exercises 59 and 60, determine
whether each statement is true or false. If a statement is
true, give a reason or cite an appropriate statement from
the text. If a statement is false, provide an example that
shows the statement is not true in all cases or cite an
appropriate statement from the text.
59. 冪2冪2 冪4 2 60. 共冪10兲2 冪100 10
61. Proof Prove that if the product of two complex numbers
is zero, then at least one of the numbers must be zero.
9781133110873_0802.qxp
3/14/12
8:23 AM
Page 399
8.2
Conjugates and Division of Complex Numbers
399
8.2 Conjugates and Division of Complex Numbers
Find the conjugate of a complex number.
Find the modulus of a complex number.
Divide complex numbers, and find the inverse of a complex matrix.
COMPLEX CONJUGATES
In Section 8.1, it was mentioned that the complex zeros of a polynomial with real
coefficients occur in conjugate pairs. For instance, in Example 5 you saw that the zeros
of p共x兲 ⫽ x 2 ⫺ 6x ⫹ 13 are 3 ⫹ 2i and 3 ⫺ 2i.
In this section, you will examine some additional properties of complex conjugates.
You will begin with the definition of the conjugate of a complex number.
Definition of the Conjugate of a Complex Number
The conjugate of the complex number z ⫽ a ⫹ bi is denoted by z and is given by
z ⫽ a ⫺ bi.
REMARK
In part (d) of Example 1, note
that 5 is its own complex
conjugate. In general, it can
be shown that a number is its
own complex conjugate if and
only if the number is real. (See
Exercise 39.)
Imaginary
axis
z = − 2 + 3i
3
−1
1
Real
axis
2
−3
Imaginary
axis
−3 −2
THEOREM 8.1
Properties of Complex Conjugates
zz ⫽ 共a ⫹ bi兲共a ⫺ bi兲 ⫽ a 2 ⫹ abi ⫺ abi ⫺ b2i 2 ⫽ a 2 ⫹ b2.
2 3
−2
−3
−4
−5
z ⫽ ⫺2 ⫺ 3i
z ⫽ 4 ⫹ 5i
z ⫽ 2i
z⫽5
PROOF
To prove the first property, let z ⫽ a ⫹ bi. Then z ⫽ a ⫺ bi and
z = 4 + 5i
5
4
3
2
1
Conjugate
z ⫽ ⫺2 ⫹ 3i
z ⫽ 4 ⫺ 5i
z ⫽ ⫺2i
z⫽5
For a complex number z ⫽ a ⫹ bi, the following properties are true.
1. zz ⫽ a2 ⫹ b2
2. zz ⱖ 0
3. zz ⫽ 0 if and only if z ⫽ 0.
4. 共z兲 ⫽ z
−2
z = − 2 − 3i
Complex Number
a.
b.
c.
d.
Geometrically, two points in the complex plane are conjugates if and only if they
are reflections in the real (horizontal) axis, as shown in Figure 8.6. Complex conjugates
have many useful properties. Some of these are shown in Theorem 8.1.
2
−4 −3
Finding the Conjugate of a Complex Number
5 6 7
Real
axis
The second and third properties follow directly from the first. Finally, the fourth
property follows from the definition of the complex conjugate. That is,
共 z 兲 ⫽ 共 a ⫹ bi 兲 ⫽ a ⫺ bi ⫽ a ⫹ bi ⫽ z.
z = 4 − 5i
Finding the Product of Complex Conjugates
Conjugate of a Complex Number
Figure 8.6
When z ⫽ 1 ⫹ 2i, you have zz ⫽ 共1 ⫺ 2i兲共1 ⫹ 2i兲 ⫽ 12 ⫹ 22 ⫽ 1 ⫹ 4 ⫽ 5.
9781133110873_0802.qxp
400
3/10/12
Chapter 8
6:57 AM
Page 400
Complex Vector Spaces
THE MODULUS OF A COMPLEX NUMBER
REMARK
The modulus of a complex
number is also called the
absolute value of the number.
In fact, when z is a real number,
ⱍzⱍ 冪a2 02 ⱍaⱍ.
Because a complex number can be represented by a vector in the complex plane, it
makes sense to talk about the length of a complex number. This length is called the
modulus of the complex number.
Definition of the Modulus of a Complex Number
ⱍⱍ
The modulus of the complex number z a bi is denoted by z and is given by
ⱍzⱍ 冪a2
b2.
Finding the Modulus of a Complex Number
For z 2 3i and w 6 i, determine the value of each modulus.
ⱍⱍ
a. z
ⱍ ⱍ
b. w
ⱍ ⱍ
c. zw
SOLUTION
a. z 冪22 32 冪13
b. w 冪6 2 共1兲2 冪37
c. Because zw 共2 3i兲共6 i兲 15 16i, you have
zw 冪15 2 16 2 冪481.
ⱍⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ ⱍ ⱍⱍ ⱍ
Note that in Example 3, zw z w . In Exercise 41, you are asked to prove that
this multiplicative property of the modulus always holds. Theorem 8.2 states that the
modulus of a complex number is related to its conjugate.
THEOREM 8.2
The Modulus of a Complex Number
ⱍⱍ
For a complex number z, z 2 zz.
PROOF
Let z a bi, then z a bi and zz 共a bi兲共a bi兲 a2 b2 z 2.
ⱍⱍ
LINEAR
ALGEBRA
APPLIED
Fractals appear in almost every part of the universe. They
have been used to study a wide variety of applications such
as bacteria cultures, the human lungs, the economy, and
galaxies. The most famous fractal is called the Mandelbrot
Set, named after the Polish-born mathematician Benoit
Mandelbrot (1924 – 2010). The Mandelbrot Set is based on
the following sequence of complex numbers.
zn 共zn1兲2 c, z1 c
The behavior of this sequence depends on the value of the
complex number c. For some values of c, the modulus of
each term zn in the sequence is less than some fixed number
N, and the sequence is bounded. This means that c is in the
Mandelbrot Set, and its point is colored black. For other
values of c, the moduli of the terms of the sequence become
infinitely large, and the sequence is unbounded. This means
that c is not in the Mandelbrot Set, and its point is assigned
a color based on “how quickly” the sequence diverges.
Andrew Park/Shutterstock.com
9781133110873_0802.qxp
3/10/12
6:57 AM
Page 401
8.2
Conjugates and Division of Complex Numbers
401
DIVISION OF COMPLEX NUMBERS
One of the most important uses of the conjugate of a complex number is in performing
division in the complex number system. To define division of complex numbers,
consider z a bi and w c di and assume that c and d are not both 0. For
the quotient
z
x yi
w
to make sense, it has to be true that
z w共x yi兲 共c di兲共x yi兲 共cx dy兲 共dx cy兲i.
But, because z a bi, you can form the linear system below.
cx dy a
dx cy b
Solving this system of linear equations for x and y yields
x
ac bd
ww
and
y
bc ad
.
ww
Now, because zw 共a bi兲共c di兲 共ac bd兲 共bc ad兲i, the following
definition is obtained.
Definition of Division of Complex Numbers
The quotient of the complex numbers z a bi and w c di is defined as
z
a bi
w
c di
ac bd bc ad
2
2
i
c d2
c d2
1
共zw 兲
w2
REMARK
If c2 d 2 0, then c d 0,
and w 0. In other words, as
is the case with real numbers,
division of complex numbers
by zero is not defined.
ⱍ ⱍ
provided c2 d 2 0.
In practice, the quotient of two complex numbers can be found by multiplying the
numerator and the denominator by the conjugate of the denominator, as follows.
a bi a bi c di
c di
c di c di
共a bi兲共c di兲
共c di兲共c di兲
共ac bd兲 共bc ad兲i
c2 d 2
ac bd bc ad
2
2
i
c d2
c d2
冢
冣
Division of Complex Numbers
冢
1
1
1
1i 1i 1
2i
2i
b.
3 4i 3 4i
a.
1i
1i 1 1
i
2
i
i
1 i2
2
2 2
2 11i
3 4i
2
11
i
3 4i
9 16
25 25
冢
冣
冣
9781133110873_0802.qxp
402
3/10/12
Chapter 8
6:57 AM
Page 402
Complex Vector Spaces
Now that you can divide complex numbers, you can find the (multiplicative)
inverse of a complex matrix, as demonstrated in Example 5.
Finding the Inverse of a Complex Matrix
Find the inverse of the matrix
A
2i
5 2i
6 2i
冤3 i
冥
and verify your solution by showing that AA 1 I2.
SOLUTION
Using the formula for the inverse of a 2
A1 6 2i
冤
ⱍ ⱍ 3 i
1
A
2 matrix from Section 2.3,
5 2i
.
2i
冥
Furthermore, because
ⱍAⱍ 共2 i兲共6 2i兲 共5 2i兲共3 i兲
共12 6i 4i 2兲 共15 6i 5i 2兲
3i
it follows that
TECHNOLOGY
If your graphing utility or
software program can perform
operations with complex
matrices, then you can verify
the result of Example 5. If you
have matrix A stored on a
graphing utility, evaluate A1.
1 6 2i 5 2i
3 i 3 i 2 i
1
1
共6 2i兲共3 i兲 共5 2i兲共3 i兲
3i 3i
共3 i兲共3 i兲 共2 i兲共3 i兲
1 20 17 i
.
10 10
7i
冤
冢
A1 冤
冥
冣冤
冥
冥
To verify your solution, multiply A and A1 as follows.
AA1 冤23 ii
5 2i 1 20
6 2i 10 10
冥 冤
1 10
17 i
10 0
7i
冥
冤
冥 冤
0
1
10
0
冥
0
1
The last theorem in this section summarizes some useful properties of complex
conjugates.
THEOREM 8.3
Properties of Complex Conjugates
For the complex numbers z and w, the following properties are true.
1. z w z w
2. z w z w
3. zw z w
4. z兾w z兾w
PROOF
To prove the first property, let z a bi and w c di. Then
z w 共a c兲 共b d兲i
共a c兲 共b d 兲i
共a bi兲 共c di兲
z w.
The proof of the second property is similar. The proofs of the other two properties are
left to you.
9781133110873_0802.qxp
3/10/12
6:57 AM
Page 403
8.2
403
Exercises
8.2 Exercises
Finding the Conjugate In Exercises 1–6, find the
complex conjugate z and geometrically represent both
z and z.
1. z 6 3i
2. z 2 5i
3. z 8i
4. z 2i
5. z 4
6. z 3
In Exercises 7–12, find the
indicated modulus, where z ⴝ 2 ⴙ i, w ⴝ ⴚ3 ⴙ 2i, and
v ⴝ ⴚ5i.
7. z
8. z2
9. zw
10. wz
11. v
12. zv 2
Finding the Modulus
ⱍⱍ
ⱍ ⱍ
ⱍⱍ
13.
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
Verify that ⱍwzⱍ ⱍwⱍⱍzⱍ ⱍzwⱍ, where z 1 i and
w 1 2i.
14. Verify that zv 2 z v 2 z v 2, where z 1 2i
and v 2 3i.
ⱍ ⱍ ⱍ ⱍⱍ ⱍ ⱍ ⱍⱍ ⱍ
In Exercises 15 – 20,
Dividing Complex Numbers
Finding the Inverse of a Complex Matrix In Exercises
31–36, determine whether the complex matrix A has an
inverse. If A is invertible, find its inverse and verify that
AAⴚ1 ⴝ I.
6 3i
2i 2 i
31. A 32. A 2i i
3
3i
冤
冥
1i
2
33. A 冤
1 1 i冥
冤
i
35. A 0
0
0
i
0
0
0
i
冥
冤
1i
34. A 冤
0
冤
1
36. A 0
0
冥
冥
2
1i
0
0
1i
0
0 1i
冥
In Exercises 37 and 38, determine
all values of the complex number z for which A is
singular. (Hint: Set det冇A冈 ⴝ 0 and solve for z.)
Singular Matrices
37. A 冤
39. Proof
冥
5
z
3i 2 i
冤
2
2i 1 i
38. A 1 i 1 i
z
1
0
0
冥
Prove that z z if and only if z is real.
perform the indicated operations.
15.
2i
i
16.
1
6 3i
17.
3 冪2i
3 冪2i
18.
5i
4i
19.
共2 i兲共3 i兲
4 2i
20.
3i
共2 i兲共5 2i兲
Operations with Complex Rational Expressions In
Exercises 21–24, perform the operation and write the
result in standard form.
21.
2
3
1i 1i
22.
2i
5
2i 2i
23.
i
2i
3i 3i
24.
1i
3
i
4i
In Exercises 25–28, use the given zero to
find all zeros of the polynomial function.
25. p共x兲 3x3 4x2 8x 8
Zero: 1 冪3i
3
2
26. p共x兲 4x 23x 34x 10
Zero: 3 i
4
3
2
27. p共x兲 x 3x 5x 21x 22 Zero: 3 冪2i
28. p共x兲 x3 4x2 14x 20
Zero: 1 3i
Finding Zeros
Powers of Complex Numbers In Exercises 29 and 30,
find each power of the complex number z.
(a) z 2
(b) z 3
(c) zⴚ1
(d) zⴚ2
29. z 2 i
30. z 1 i
1i
.
6 2i
(a) Without performing any calculations, describe
how to find the quotient.
(b) Explain why the process described in part (a)
results in a complex number of the form a bi.
(c) Find the quotient.
40.
Consider the quotient
41. Proof Prove that for any two complex numbers z and
w, each of the statements below is true.
(a) zw z w
(b) If w 0, then z兾w z 兾 w .
42. Graphical Interpretation Describe the set of
points in the complex plane that satisfies each of the
statements below.
(a) z 3
(b) z 1 i 5
(c) z i 2
(d) 2 z 5
n
43. (a) Evaluate 共1兾i兲 for n 1, 2, 3, 4, and 5.
(b) Calculate 共1兾i兲2000 and 共1兾i兲2010.
(c) Find a general formula for 共1兾i兲n for any positive
integer n.
1i 2
44. (a) Verify that
i.
冪2
(b) Find the two square roots of i.
(c) Find all zeros of the polynomial x 4 1.
ⱍ ⱍ ⱍ ⱍⱍ ⱍ
ⱍ ⱍ ⱍⱍⱍ ⱍ
ⱍⱍ
ⱍ ⱍ
ⱍ
冢
冣
ⱍⱍ
ⱍ
9781133110873_0803.qxp
404
3/10/12
Chapter 8
6:54 AM
Page 404
Complex Vector Spaces
8.3 Polar Form and DeMoivre’s Theorem
Determine the polar form of a complex number, convert between
the polar form and standard form of a complex number, and
multiply and divide complex numbers in polar form.
Use DeMoivre’s Theorem to find powers and roots of complex
numbers in polar form.
POLAR FORM OF A COMPLEX NUMBER
Imaginary
axis
At this point you can add, subtract, multiply, and divide complex numbers. However,
there is still one basic procedure that is missing from the algebra of complex numbers.
To see this, consider the problem of finding the square root of a complex number such
as i. When you use the four basic operations (addition, subtraction, multiplication, and
division), there seems to be no reason to guess that
(a, b)
r
b
冪i θ
a
0
Real
axis
That is,
1i
Standard Form: a + bi
Polar Form: r(cos θ + i sin θ )
Figure 8.7
1i
.
冪2
冢 冪2 冣 i.
2
To work effectively with powers and roots of complex numbers, it is helpful to use a
polar representation for complex numbers, as shown in Figure 8.7. Specifically, if
a bi is a nonzero complex number, then let be the angle from the positive real axis
to the radial line passing through the point 共a, b兲 and let r be the modulus of a bi.
This leads to the following.
a r cos b r sin r 冪a 2 b2
REMARK
The polar form of z 0
is expressed as
z 0共cos i sin 兲, where
is any angle.
So, a bi 共r cos 兲 共r sin 兲i, from which the polar form of a complex number
is obtained.
Definition of the Polar Form of a Complex Number
The polar form of the nonzero complex number z a bi is given by
z r共cos i sin 兲
where a r cos , b r sin , r 冪a 2 b2, and tan b兾a. The number r
is the modulus of z and is the argument of z.
Because there are infinitely many choices for the argument, the polar form of a
complex number is not unique. Normally, the values of that lie between and are used, although on occasion it is convenient to use other values. The value of that
satisfies the inequality
< Principal argument
is called the principal argument and is denoted by Arg(z). Two nonzero complex
numbers in polar form are equal if and only if they have the same modulus and the
same principal argument.
9781133110873_0803.qxp
3/10/12
6:54 AM
Page 405
8.3
Polar Form and DeMoivre’s Theorem
405
Finding the Polar Form of a Complex Number
Find the polar form of each of the complex numbers. (Use the principal argument.)
a. z 1 i
b. z 2 3i
c. z i
SOLUTION
a. Because a 1 and b 1, then r 2 1 2 共1兲2 2, which implies that
r 冪2. From a r cos and b r sin ,
cos 冪2
a
1
冪2
r
2
and
sin 冪2
b
1
.
冪2
r
2
So, 兾4 and
冤 冢 4 冣 i sin冢 4 冣冥.
z 冪2 cos b. Because a 2 and b 3, then r 2 2 2 32 13, which implies that r 冪13.
So,
cos a
2
冪13
r
and
sin b
3
冪13
r
and it follows that ⬇ 0.98. So, the polar form is
冤
冥
z ⬇ 冪13 cos 共0.98兲 i sin共0.98兲 .
c. Because a 0 and b 1, it follows that r 1 and 兾2, so
冢
z 1 cos
i sin .
2
2
冣
The polar forms derived in parts (a), (b), and (c) are depicted graphically in Figure 8.8.
Imaginary
axis
Imaginary
axis
−2
−1
2
4
1
3
θ
−1
2
Real
axis
2
1
z=1−i
−2
冤 冢 4 冣 i sin 冢 4 冣冥
Imaginary
axis
z=i
θ
Real
axis
1
冢
c. z 1 cos
Figure 8.8
θ
1
a. z 冪2 cos 1
z = 2 + 3i
i sin
2
2
冣
2
Real
axis
b. z ⬇ 冪13 关cos共0.98兲 i sin共0.98兲兴
9781133110873_0803.qxp
406
3/10/12
Chapter 8
6:54 AM
Page 406
Complex Vector Spaces
Converting from Polar to Standard Form
Express the complex number in standard form.
冤 冢 3 冣 i sin冢 3 冣冥
z 8 cos SOLUTION
Because cos共 兾3兲 1兾2 and sin共兾3兲 冪3兾2, obtain the standard form
冤 冢 3 冣 i sin冢 3 冣冥 8冤12 i
z 8 cos 冪3
2
冥
4 4冪3i.
The polar form adapts nicely to multiplication and division of complex numbers.
Suppose you have two complex numbers in polar form
z1 r1共cos 1 i sin 1兲 and
z2 r2共cos 2 i sin 2 兲.
Then the product of z1 and z2 is expressed as
z1 z 2 r1 r2共cos 1 i sin 1兲共cos 2 i sin 2兲
r1r2 关共cos 1 cos 2 sin 1 sin 2兲 i 共cos 1 sin 2 sin 1 cos 2兲兴.
Using the trigonometric identities cos共1 2兲 cos 1 cos 2 sin 1 sin 2 and
sin共1 2兲 sin 1 cos 2 cos 1 sin 2 , you have
z1z 2 r1r2 关cos共1 2兲 i sin共1 2兲兴.
This establishes the first part of the next theorem. The proof of the second part is left to
you. (See Exercise 75.)
THEOREM 8.4
Product and Quotient of Two Complex Numbers
Given two complex numbers in polar form
z1 r1共cos 1 i sin 1兲 and
z2 r2共cos 2 i sin 2 兲
the product and quotient of the numbers are as follows.
z1z2 r1r2 关cos共1 2兲 i sin共1 2兲兴
Product
z1 r1
z 2 r2 关cos共1 2兲 i sin共1 2兲兴, z2 0
Quotient
LINEAR
ALGEBRA
APPLIED
Elliptic curves are the foundation for elliptic curve
cryptography (ECC), a type of public key cryptography for
secure communications over the Internet. ECC has gained
popularity due to its computational and bandwidth
advantages over traditional public key algorithms.
One specific variety of elliptic curve is formed using
Eisenstein integers. Eisenstein integers are complex
1 冪3
numbers of the form z a b, where i,
2
2
and a and b are integers. These numbers can be graphed
as intersection points of a triangular lattice in the complex
plane. Dividing the complex plane by the lattice of all
Eisenstein integers results in an elliptic curve.
thumb/Shutterstock.com
9781133110873_0803.qxp
3/10/12
6:54 AM
Page 407
8.3
407
Polar Form and DeMoivre’s Theorem
Theorem 8.4 says that to multiply two complex numbers in polar form, multiply
moduli and add arguments. To divide two complex numbers, divide moduli and subtract
arguments. (See Figure 8.9.)
Imaginary
axis
Imaginary
axis
z1
z2
z1z2
θ1 + θ 2 r
2
r1r2
z2
r2
z1
θ 2 r1
θ1
r1
θ1
θ2
z1
z2
r1
r2
θ1 − θ 2
Real
axis
Real
axis
To multiply z1 and z2 :
Multiply moduli and add arguments.
To divide z1 and z2 :
Divide moduli and subtract arguments.
Figure 8.9
Multiplying and Dividing in Polar Form
Find z1z2 and z1兾z2 for the complex numbers
冢
z1 5 cos
i sin
4
4
冣
and z2 1
cos i sin .
3
6
6
冢
冣
SOLUTION
Because z1 and z2 are in polar form, apply Theorem 8.4, as follows.
multiply
z1z2 共5兲
5
5
冢3冣 冤cos冢 4 6 冣 i sin冢 4 6 冣冥 3 冢cos 12 i sin 12 冣
1
add
5
add
divide
z1
5
cos
i sin
z2 1兾3
4
6
4
6
冤 冢
冣
冢
subtract
冣冥 15冢cos 12 i sin 12冣
subtract
Use the standard forms of z1 and z2 to check the multiplication in Example 3.
For instance,
z1z2 冢5 2 2 5 2 2 i冣冢 63 16 i冣 5126 5122 i 5126 i 5122 i
冪
冪
冪
冪
冪
冪
冪
2
5冪6 5冪2 5冪6 5冪2
i
12
12
5 冪6 冪2 冪6 冪2
i .
3
4
4
冢
REMARK
Try verifying the division in
Example 3 using the standard
forms of z1 and z2.
冣
To verify that this answer is equivalent to the result in Example 3, use the formulas for
cos 共u v兲 and sin 共u v兲 to obtain
cos
冢512冣 cos冢6 4 冣 冪6 冪2
4
and sin
冢512冣 sin冢6 4 冣 冪6 冪2
4
.
9781133110873_0803.qxp
408
3/10/12
Chapter 8
6:54 AM
Page 408
Complex Vector Spaces
DEMOIVRE’S THEOREM
The final topic in this section involves procedures for finding powers and roots of
complex numbers. Repeated use of multiplication in the polar form yields
z r 共cos i sin 兲
z 2 r 共cos i sin 兲r 共cos i sin 兲
r 2共cos 2 i sin 2兲
and
z3 r 共cos i sin 兲r 2 共cos 2 i sin 2兲
r 3共cos 3 i sin 3兲.
Similarly,
z4 r 4共cos 4 i sin 4兲
and
z 5 r 5共cos 5 i sin 5兲.
This pattern leads to the next important theorem, named after the French mathematician
Abraham DeMoivre (1667–1754). You are asked to prove this theorem in Review
Exercise 85.
THEOREM 8.5
DeMoivre’s Theorem
If z r 共cos i sin 兲 and n is any positive integer, then
zn r n共cos n i sin n兲.
Raising a Complex Number
to an Integer Power
Find 共1 冪3 i兲 and write the result in standard form.
12
SOLUTION
First convert to polar form. For 1 冪3i,
r 冪共1兲2 共冪3 兲2 2
and tan 冪3
1
冪3
which implies that 2兾3. So,
冢
1 冪3i 2 cos
2
2
.
i sin
3
3
冣
By DeMoivre’s Theorem,
2
2 12
i sin
3
3
12共2兲
12共2兲
212 cos
i sin
3
3
4096共cos 8 i sin 8兲
4096 关1 i 共0兲兴
4096.
共1 冪3i兲12 冤 2冢cos
冤
冣冥
冥
9781133110873_0803.qxp
3/10/12
6:54 AM
Page 409
8.3
Polar Form and DeMoivre’s Theorem
409
Recall that a consequence of the Fundamental Theorem of Algebra is that a
polynomial of degree n has n zeros in the complex number system. So, a polynomial
such as p共x兲 x6 1 has six zeros, and in this case you can find the six zeros by
factoring and using the Quadratic Formula.
x6 1 共x3 1兲共x3 1兲
共x 1兲共x 2 x 1兲共x 1兲共x 2 x 1兲
Consequently, the zeros are
x ± 1,
x
1 ± 冪3i
,
2
and x 1 ± 冪3i
.
2
Each of these numbers is called a sixth root of 1. In general, the nth root of a complex
number is defined as follows.
Definition of the nth Root of a Complex Number
The complex number w a bi is an nth root of the complex number z when
z w n 共a bi兲n.
DeMoivre’s Theorem is useful in determining roots of complex numbers. To see
how this is done, let w be an nth root of z, where
w s共cos i sin 兲 and z r 共cos i sin 兲.
Then, by DeMoivre’s Theorem
w n s n共cos n
i sin n
兲
and because w n z, it follows that
s n 共cos n
i sin n
兲 r 共cos i sin 兲.
Now, because the right and left sides of this equation represent equal complex numbers,
n r, and equate principal
equate moduli to obtain s n r, which implies that s 冪
arguments to conclude that and n
must differ by a multiple of 2. Note that r is a
n r is also a positive real number. Consequently, for
positive real number and so s 冪
some integer k, n
2k, which implies that
REMARK
Note that when k exceeds
n 1, the roots begin to
repeat. For instance, when
k n, the angle is
2n
2
n
n
which yields the same values
for the sine and cosine as
k 0.
2 k
.
n
Finally, substituting this value of into the polar form of w produces the result stated
in the next theorem.
THEOREM 8.6
The nth Roots of a Complex Number
For any positive integer n, the complex number z r 共cos i sin 兲 has
exactly n distinct roots. These n roots are given by
2 k
2k
i sin
n
n
冤 冢
n r cos
冪
冣
where k 0, 1, 2, . . . , n 1.
冢
冣冥
9781133110873_0803.qxp
410
3/10/12
Chapter 8
6:54 AM
Complex Vector Spaces
Imaginary
axis
n
2π
n
r
2π
n
Page 410
Real
axis
The nth Roots of a
Complex Number
Figure 8.10
The formula for the nth roots of a complex
number has a nice geometric interpretation, as
shown in Figure 8.10. Because the nth roots all
n r, they lie on a
have the same modulus (length) 冪
n
circle of radius 冪r with center at the origin.
Furthermore, the n roots are equally spaced
around the circle, because successive nth roots
have arguments that differ by 2兾n.
You have already found the sixth roots of 1 by
factoring and the Quadratic Formula. Try solving
the same problem using Theorem 8.6 to get the
roots shown in Figure 8.11. When Theorem 8.6 is
applied to the real number 1, the nth roots have a
special name—the nth roots of unity.
Imaginary
axis
−1 + 3i
2
2
−1
1+ 3
i
2
2
1
Real
axis
1− 3
−1 − 3i
i
2
2
2
2
The Sixth Roots of Unity
Figure 8.11
Finding the nth Roots of a Complex Number
Determine the fourth roots of i.
SOLUTION
In polar form,
冢
i 1 cos
i sin
2
2
冣
so r 1 and 兾2. Then, by applying Theorem 8.6,
冤 冢4兾2 2k4冣 i sin冢4兾2 2k4冣冥
k
k
cos冢 冣 i sin冢 冣.
8
2
8
2
4 1 cos
i1兾4 冪
Setting k 0, 1, 2, and 3,
i sin
8
8
5
5
z 2 cos
i sin
8
8
9
9
z3 cos
i sin
8
8
13
13
z4 cos
i sin
8
8
z1 cos
REMARK
In Figure 8.12, note that when
each of the four angles
兾8, 5兾8, 9兾8, and 13兾8 is
multiplied by 4, the result is of
the form 共兾2兲 2k.
as shown in Figure 8.12.
cos 5π + i sin 5π
8
8
Imaginary
axis
cos π + i sin π
8
8
Real
axis
cos 9π + i sin 9π
8
8
cos 13π + i sin 13π
8
8
Figure 8.12
9781133110873_0803.qxp
3/10/12
6:54 AM
Page 411
8.3
411
Exercises
8.3 Exercises
In Exercises 1–4, express
the complex number in polar form.
Imaginary
1. Imaginary
2.
Converting to Polar Form
axis
axis
1
Real
axis
2
1 + 3i
3
2
−1
1
−2
2 − 2i
3.
Imaginary
axis
−1
4.
−6
1
Real
axis
2
3
3i
2
Real
axis
− 6 −5 −4 −3 − 2
−2
−3
1
−1
Real
axis
1
Graphing and Converting to Polar Form In Exercises
5–16, represent the complex number graphically, and
give the polar form of the number. (Use the principal
argument.)
5. 2 2i
6. 2 2i
7. 2共1 冪3i 兲
8. 52 共冪3 i 兲
9. 6i
10. 2i
11. 7
12. 4
13. 3 冪3i
14. 2冪2 i
15. 1 2i
16. 5 2i
Graphing and Converting to Standard Form In
Exercises 17–26, represent the complex number
graphically, and give the standard form of the number.
17. 2 cos i sin
2
2
冢
冣
3
3
18. 5 cos
i sin
4
4
冢
冣
3
5
5
19.
cos
i sin
2
3
3
冢
20.
3
7
7
cos
i sin
4
4
4
21.
15
cos i sin
4
4
4
冢
冢
冢
冢
i sin
6
6
冢
5
5
i sin
6
6
冣
22. 8 cos
3
3
i sin
2
2
24. 6 cos
25. 7共cos 0 i sin 0兲
冣
冣
26. 9共cos i sin 兲
冣
28.
冤 4 冢cos 2 i sin 2 冣冥冤6 冢cos 4 i sin 4 冣冥
3
29. 关0.5共cos i sin 兲兴关 0.5共cos 关兴 i sin关兴兲兴
30.
冤3冢cos 3 i sin 3 冣冥冤 3 冢cos
31.
2关cos共2兾3兲 i sin共2兾3兲兴
4[cos共5兾6兲 i sin共5兾6兲]
32.
cos共5兾3兲 i sin共5兾3兲
cos i sin
1
2
2
i sin
3
3
冣冥
12关cos共兾3兲 i sin共兾3兲兴
3关cos共兾6兲 i sin共兾6兲兴
9关cos共3兾4兲 i sin共3兾4兲兴
34.
5关cos共 兾4兲 i sin共 兾4兲兴
33.
In Exercises
35–44, use DeMoivre’s Theorem to find the indicated
powers of the complex number. Express the result in
standard form.
35. 共1 i兲 4
36. 共2 2i兲 6
37. 共1 i兲 10
38. 共冪3 i 兲 7
3
39. 共1 冪3i 兲
Finding Powers of Complex Numbers
40.
冤5冢cos 9 i sin 9 冣冥
41.
冤3冢cos
42.
冢cos
43.
冤2冢cos 2 i sin 2 冣冥
5
5
i sin
6
6
5
5
i sin
4
4
44. 5 cos
冣
冤3冢cos 3 i sin 3 冣冥冤4冢cos 6 i sin 6 冣冥
冤冢
冣
27.
Imaginary
axis
3
2
1
23. 4 cos
In Exercises
27–34, perform the indicated operation and leave the
result in polar form.
Multiplying and Dividing in Polar Form
冣
3
冣冥
4
10
3
3
i sin
2
2
8
冣冥
4
Finding Square Roots of a Complex Number In
Exercises 45–52, find the square roots of the complex
number.
45. 2i
46. 5i
47. 3i
48. 6i
49. 2 2i
50. 2 2i
冪
51. 1 3i
52. 1 冪3i
9781133110873_0803.qxp
412
3/10/12
Chapter 8
6:54 AM
Page 412
Complex Vector Spaces
Finding and Graphing nth Roots
In Exercises 53–64,
(a) use Theorem 8.6 to find the indicated roots, (b)
represent each of the roots graphically, and (c) express
each of the roots in standard form.
冢
53. Square roots: 16 cos
冢
54. Square roots: 9 cos
冢
冢
57.
58.
59.
60.
61.
62.
63.
64.
冣
2
2
i sin
3
3
55. Fourth roots: 16 cos
56. Fifth roots: 32 cos
i sin
3
3
5
5
i sin
6
6
76. Proof Show that the complex conjugate of
z r共cos i sin 兲 is
z r 关cos共 兲 i sin共 兲兴.
77. Use the polar forms of z and z in Exercise 76 to find
each of the following.
(a) zz
(b) z兾z, z 0
78. Proof Show that the negative of z r共cos i sin 兲
is
冣
4
4
i sin
3
3
冣
冣
Square roots: 25i
Fourth roots: 625i
Cube roots: 125
2 共1 冪3i兲
Cube roots: 4冪2 共1 i兲
Cube roots: 8
Fourth roots: 81i
Fourth roots: 1
Cube roots: 1000
z r 关cos共 兲 i sin共 兲兴.
79. Writing
In Exercises 65–72,
find all the solutions of the equation and represent your
solutions graphically.
65. x 4 256i 0
66. x 4 16i 0
67. x 3 1 0
68. x3 27 0
69. x 5 243 0
70. x 4 81 0
71. x 3 64i 0
72. x 4 i 0
73. Electrical Engineering In an electric circuit, the
formula V I Z relates voltage drop V, current I, and
impedance Z, where complex numbers can represent
each of these quantities. Find the impedance when the
voltage drop is 5 5i and the current is 2 4i.
Use the graph of the roots of a
complex number.
(a) Write each of the roots in trigonometric form.
(b) Identify the complex number whose roots are
given. Use a graphing utility to verify your results.
Imaginary
Imaginary
(i)
(ii)
axis
30°
2
axis
2
3
30°
2 1
−1
Real
axis
i sin .
6
6
Sketch z, iz, and z兾i in the complex plane.
(b) What is the geometric effect of multiplying a
complex number z by i? What is the geometric
effect of dividing z by i?
80. Calculus Recall that the Maclaurin series for e x,
sin x, and cos x are
x 2 x3 x 4 . . .
ex 1 x 2!
3!
4!
(a) Let
Finding and Graphing Solutions
74.
75. Proof When provided with two complex numbers
z 1 r1共cos 1 i sin 1兲 and z 2 r2共cos 2 i sin 2 兲,
with z 2 0, prove that
r1
z1
z 2 r 2 关cos共 1 2兲 i sin共 1 2兲兴.
3
45°
45°
45°
45°
3
3
Real
axis
冢
z r共cos i sin 兲 2 cos
sin x x 冣
x 3 x5 x 7 . . .
3!
5!
7!
x 2 x 4 x6 . . .
.
2!
4!
6!
Substitute x i in the series for ex and show that
ei cos i sin .
Show that any complex number z a bi can be
expressed in polar form as z rei.
Prove that if z rei, then z rei.
Prove the formula ei 1.
cos x 1 (a)
(b)
(c)
(d)
True or False? In Exercises 81 and 82, determine
whether each statement is true or false. If a statement is
true, give a reason or cite an appropriate statement from
the text. If a statement is false, provide an example that
shows the statement is not true in all cases or cite an
appropriate statement from the text.
81. Although the square of the complex number bi is given
by 共bi兲2 b2, the absolute value of the complex
number z a bi is defined as a bi 冪a2 b2.
82. Geometrically, the nth roots of any complex number z
are all equally spaced around the unit circle centered at
the origin.
ⱍ
ⱍ
9781133110873_0804.qxp
3/10/12
6:56 AM
Page 413
8.4
Complex Vector Spaces and Inner Products
413
8.4 Complex Vector Spaces and Inner Products
Recognize and perform vector operations in complex vector spaces
C n, represent a vector in C n by a basis, and find the Euclidean
inner product, Euclidean norm, and Euclidean distance in C n.
Recognize complex inner product spaces.
COMPLEX VECTOR SPACES
All the vector spaces studied so far in the text have been real vector spaces because the
scalars have been real numbers. A complex vector space is one in which the scalars are
complex numbers. So, if v1, v2, . . . , vm are vectors in a complex vector space, then a
linear combination is of the form
c1v1 c 2v2 cmvm
where the scalars c1, c2, . . . , cm are complex numbers. The complex version of Rn is
the complex vector space C n consisting of ordered n-tuples of complex numbers. So,
a vector in C n has the form
v 共a1 b1i, a 2 b 2i, . . . , an bni 兲.
It is also convenient to represent vectors in C n by column matrices of the form
冤 冥
a1 b1i
a bi
v 2 .. 2 .
.
an bni
As with R n, the operations of addition and scalar multiplication in C n are performed
component by component.
Vector Operations in C n
Let
v 共1 2i, 3 i兲 and u 共2 i, 4兲
be vectors in the complex vector space C 2. Determine each vector.
a. v u
b. 共2 i兲v
c. 3v 共5 i兲u
SOLUTION
a. In column matrix form, the sum v u is
1 2i
2 i
1 3i
vu
.
3i
4
7i
冤
冥 冤
冥 冤
冥
b. Because 共2 i兲共1 2i兲 5i and 共2 i兲共3 i兲 7 i,
共2 i 兲v 共2 i 兲共1 2i, 3 i 兲
共5i, 7 i 兲.
c. 3v 共5 i兲u 3共1 2i, 3 i兲 共5 i兲共2 i, 4兲
共3 6i, 9 3i兲 共9 7i, 20 4i兲
共12 i, 11 i兲
9781133110873_0804.qxp
414
3/10/12
Chapter 8
6:56 AM
Page 414
Complex Vector Spaces
Many of the properties of R n are shared by C n. For instance, the scalar multiplicative
identity is the scalar 1 and the additive identity in C n is 0 共0, 0, 0, . . . , 0兲. The
standard basis for C n is simply
e1 共1, 0, 0, . . . , 0兲
e 2 共0, 1, 0, . . . , 0兲
.
.
.
en 共0, 0, 0, . . . , 1兲
which is the standard basis for Rn. Because this basis contains n vectors, it follows that
the dimension of C n is n. Other bases exist; in fact, any linearly independent set of n
vectors in C n can be used, as demonstrated in Example 2.
Verifying a Basis
Show that S 再v1, v2, v3冎 再共i, 0, 0兲, 共i, i, 0兲, 共0, 0, i兲冎 is a basis for C 3.
SOLUTION
Because the dimension of C 3 is 3, the set 再v1, v2, v3冎 will be a basis if it is linearly
independent. To check for linear independence, set a linear combination of the vectors
in S equal to 0, as follows.
c1v1 c2v2 c3v3 共0, 0, 0兲
共c1i, 0, 0兲 共c2i, c2i, 0兲 共0, 0, c3i兲 共0, 0, 0兲
共共c1 c2 兲i, c2i, c3i兲 共0, 0, 0兲
This implies that
共c1 c 2 兲i 0
c2i 0
c3i 0.
So, c1 c2 c3 0, and 再v1, v2, v3冎 is linearly independent.
Representing a Vector in C n by a Basis
Use the basis S in Example 2 to represent the vector v 共2, i, 2 i兲.
SOLUTION
By writing
v c1v1 c2v2 c3v3
共共c1 c 2 兲 i, c2i, c3i兲
共2, i, 2 i兲
you can obtain
共c1 c 2 兲 i 2
c2i i
c3i 2 i
REMARK
Try verifying that this linear
combination yields 共2, i, 2 i 兲.
which implies that
c2 1,
c1 2i
1 2i,
i
and
So, v 共1 2i兲v1 v2 共1 2i兲v3.
c3 2i
1 2i.
i
9781133110873_0804.qxp
3/10/12
6:56 AM
Page 415
8.4
Complex Vector Spaces and Inner Products
415
Other than C n, there are several additional examples of complex vector spaces.
For instance, the set of m n complex matrices with matrix addition and scalar
multiplication forms a complex vector space. Example 4 describes a complex vector
space in which the vectors are functions.
The Space of Complex-Valued Functions
Consider the set S of complex-valued functions of the form
f共x兲 f1共x兲 if 2共x兲
where f1 and f2 are real-valued functions of a real variable. The set of complex numbers
forms the scalars for S, and vector addition is defined by
f共x兲 g共x兲 关f1共x兲 if 2共x兲兴 关g1(x兲 i g2共x兲兴
关f1共x兲 g1共x兲兴 i 关f 2共x兲 g2共x兲兴.
It can be shown that S, scalar multiplication, and vector addition form a complex
vector space. For instance, to show that S is closed under scalar multiplication, let
c a bi be a complex number. Then
cf共x兲 共a bi兲关f1共x兲 if 2共x兲兴
关af1共x兲 bf 2共x兲兴 i 关bf1共x兲 af 2共x兲兴
is in S.
REMARK
Note that if u and v happen to
be “real,” then this definition
agrees with the standard inner
(or dot) product in Rn.
The definition of the Euclidean inner product in C n is similar to the standard dot
product in R n, except that here the second factor in each term is a complex conjugate.
Definition of the Euclidean Inner Product in C n
Let u and v be vectors in C n. The Euclidean inner product of u and v is given by
u v u1v1 u2v2 unvn.
Finding the Euclidean Inner Product in C 3
Determine the Euclidean inner product of the vectors
u 共2 i, 0, 4 5i兲 and v 共1 i, 2 i, 0兲.
SOLUTION
u v u1v1 u2v2 u 3v3
共2 i兲共1 i兲 0共2 i兲 共4 5i兲共0兲
3i
Several properties of the Euclidean inner product C n are stated in the following theorem.
THEOREM 8.7
Properties of the Euclidean Inner Product
Let u, v, and w be vectors in C n and let k be a complex number. Then the
following properties are true.
1. u v v u
2. 共u v兲 w u w v w
3. 共ku兲 v k共u v兲
4. u 共kv兲 k共u v兲
5. u u 0
6. u u 0 if and only if u 0.
9781133110873_0804.qxp
416
3/10/12
Chapter 8
6:56 AM
Page 416
Complex Vector Spaces
PROOF
The proof of the first property is shown below, and the proofs of the remaining
properties have been left to you (see Exercises 59–63). Let
u 共u1, u 2, . . . , un 兲 and v 共v1, v2, . . . , vn 兲.
Then
v
u v1u1 v2u2
. . . vn un
v1u1 v2u2 . . . vnun
v1u1 v2u 2 . . . vnun
u1v1 u2v2 unvn
u v.
The Euclidean inner product in C n is used to define the Euclidean norm (or length)
of a vector in C n and the Euclidean distance between two vectors in C n.
Definitions of the Euclidean Norm and Distance in C n
The Euclidean norm (or length) of u in C n is denoted by 储u储 and is
储u储 共u u兲1兾2.
The Euclidean distance between u and v is
d共u, v兲 储u v储 .
The Euclidean norm and distance may be expressed in terms of components, as
follows (see Exercise 51).
ⱍ ⱍ
ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ
储u储 共 u1 2 u2 2 . . . u n 2兲1兾2
d共u, v兲 共 u1 v1 2 u 2 v2 2 . . . un vn 2兲1兾2
ⱍ
ⱍ
Finding the Euclidean Norm and Distance in C n
Let u 共2 i, 0, 4 5i兲 and v 共1 i, 2 i, 0兲.
a. Find the norms of u and v.
b. Find the distance between u and v.
SOLUTION
a. 储u储 共 u1 2 u 2 2 u 3 2兲1兾2
关共22 12兲 共02 02兲 共42 共5兲 2兲兴1兾2
共5 0 41兲1兾2
冪46
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
储v储 共 v1 2 v2 2 v3 2兲1兾2
关共12 12兲 共22 12兲 共02 02兲兴1兾2
共2 5 0兲1兾2
冪7
b. d共u, v兲 储u v储
储共1, 2 i, 4 5i兲储
关共12 02兲 共共2兲2 共1兲2兲 共42 共5兲2兲兴1兾2
共1 5 41兲1兾2
冪47
9781133110873_0804.qxp
3/10/12
6:56 AM
Page 417
8.4
Complex Vector Spaces and Inner Products
417
COMPLEX INNER PRODUCT SPACES
The Euclidean inner product is the most commonly used inner product in C n. On occasion,
however, it is useful to consider other inner products. To generalize the notion of an
inner product, use the properties listed in Theorem 8.7.
Definition of a Complex Inner Product
Let u and v be vectors in a complex vector space. A function that associates u
and v with the complex number 具u, v典 is called a complex inner product when
it satisfies the following properties.
1. 具u, v典 具v, u典
2. 具u v, w典 具u, w典 具v, w典
3. 具ku, v典 k具u, v典
4. 具u, u典 0 and 具u, u典 0 if and only if u 0.
A complex vector space with a complex inner product is called a complex inner
product space or unitary space.
A Complex Inner Product Space
Let u 共u1, u2兲 and v 共v1, v2兲 be vectors in the complex space C 2. Show that the
function defined by
具u, v典 u1v1 2u2v2
is a complex inner product.
SOLUTION
Verify the four properties of a complex inner product, as follows.
1. 具v, u典 v1u1 2v2u2 u1v1 2u2v2 具u, v典
2. 具u v, w典 共u1 v1兲 w1 2共u2 v2兲w2
共u1w1 2u 2w2 兲 共v1w1 2v2w2 兲
具u, w典 具v, w典
3. 具ku, v典 共ku1兲v1 2共ku 2 兲v2 k 共u1v1 2u2v2 兲 k 具u, v典
4. 具u, u典 u1u1 2u2 u2 u1 2 2 u2 2 0
Moreover, 具u, u典 0 if and only if u1 u2 0.
ⱍ ⱍ
ⱍ ⱍ
Because all the properties hold, 具u, v典 is a complex inner product.
LINEAR
ALGEBRA
APPLIED
Complex vector spaces and inner products have an important
application called the Fourier transform, which decomposes
a function into a sum of orthogonal basis functions. The given
function is projected onto the standard basis functions
for varying frequencies to get the Fourier amplitudes for
each frequency. Like Fourier coefficients and the Fourier
approximation, this transform is named after the French
mathematician Jean-Baptiste Joseph Fourier (1768 –1830).
The Fourier transform is integral to the study of signal
processing. To understand the basic premise of this
transform, imagine striking two piano keys simultaneously.
Your ear receives only one signal, the mixed sound of the
two notes, and yet your brain is able to separate the notes.
The Fourier transform gives a mathematical way to take a
signal and separate out its frequency components.
Eliks/Shutterstock.com
9781133110873_0804.qxp
418
3/10/12
Chapter 8
6:56 AM
Page 418
Complex Vector Spaces
8.4 Exercises
Vector Operations In Exercises 1– 8, perform the
indicated operation using
u ⴝ 冇i, 3 ⴚ i冈, v ⴝ 冇2 ⴙ i, 3 ⴙ i兲, and w ⴝ 冇4i, 6冈.
1. 3u
2. 4iw
3. 共1 2i兲w
4. iv 3w
5. u 共2 i兲v
6. 共6 3i兲v 共2 2i兲w
7. u iv 2iw
8. 2iv 共3 i兲w u
Finding the Euclidean Norm
Linear Dependence or Independence In Exercises
9–12, determine whether the set of vectors is linearly
independent or linearly dependent.
9. 再共1, i兲, 共i, 1兲冎
10. 再共1 i, 1 i, 1兲, 共i, 0, 1兲, 共2, 1 i, 0兲冎
11. 再共1, i, 1 i兲, 共0, i, i兲, 共0, 0, 1兲冎
12. 再共1 i, 1 i, 0兲, 共1 i, 0, 0兲, 共0, 1, 1兲冎
Finding the Euclidean Distance In Exercises 35–40,
determine the Euclidean distance between u and v.
35. u 共1, 0兲, v 共i, i兲
36. u 共2 i, 4, i兲, v 共2 i, 4, i兲
37. u 共i, 2i, 3i兲, v 共0, 1, 0兲
38. u 共冪2, 2i, i兲, v 共i, i, i兲
39. u 共1, 0兲, v 共0, 1兲
40. u 共1, 2, 1, 2i兲, v 共i, 2i, i, 2兲
In Exercises 13 –16, determine
whether S is a basis for C n.
13. S 再共1, i兲, 共i, 1兲冎
14. S 再共1, i兲, 共i, 1兲冎
15. S 再共i, 0, 0兲, 共0, i, i兲, 共0, 0, 1兲冎
16. S 再共1 i, 0, 1兲, 共2, i, 1 i兲, 共1 i, 1, 1兲冎
Verifying a Basis
In Exercises 17–20,
express v as a linear combination of each of the following
basis vectors.
(a) {冇i, 0, 0冈, 冇i, i, 0冈, 冇i, i, i冈}
(b) {冇1, 0, 0冈, 冇1, 1, 0冈, 冇0, 0, 1 ⴙ i冈冎
17. v 共1, 2, 0兲
18. v 共1 i, 1 i, 3兲
19. v 共i, 2 i, 1兲
20. v 共i, i, i兲
Representing a Vector by a Basis
Finding Euclidean Inner Products In Exercises 21 and
22, determine the Euclidean inner product u v.
21. u 共i, 2i, 1 i兲
22. u 共4 i, i, 0兲
v 共3i, 0, 1 2i兲
v 共1 3i, 2, 1 i兲
In Exercises
23–26, let u ⴝ 冇1 ⴚ i, 3i兲, v ⴝ 冇2i, 2 ⴙ i冈, w ⴝ 冇1 ⴙ i, 0冈,
and k ⴝ ⴚi. Evaluate the expressions in parts (a) and (b)
to verify that they are equal.
23. (a) u v
24. (a) 共u v兲 w
(b) v u
(b) u w v w
25. (a) 共ku兲 v
26. (a) u 共kv兲
(b) k共u v兲
(b) k 共u v兲
Properties of Euclidean Inner Products
In Exercises 27–34,
determine the Euclidean norm of v.
27. v 共i, i兲
28. v 共1, 0兲
29. v 3共6 i, 2 i兲
30. v 共2 3i, 2 3i兲
31. v 共1, 2 i, i兲
32. v 共0, 0, 0兲
33. v 共1 2i, i, 3i, 1 i兲
34. v 共2, 1 i, 2 i, 4i兲
In Exercises 41–44, determine
whether the function is a complex inner product, where
u ⴝ 冇u1, u2冈 and v ⴝ 冇v1, v2冈.
41. 具u, v典 u1 u2v2
42. 具u, v典 共u1 v1兲 2共u2 v2兲
43. 具u, v典 4u1v1 6u2v2
44. 具u, v典 u1v1 u2v2
Complex Inner Products
In Exercises 45–48,
use the inner product 具u, v典 ⴝ u1v1 ⴙ 2u2v2 to find
具u, v典.
45. u 共2i, i兲 and v 共i, 4i兲
46. u 共3 i, i兲 and v 共2 i, 2i兲
47. u 共2 i, 2 i兲 and v 共3 i, 3 2i兲
48. u 共4 2i, 3兲 and v 共2 3i, 2兲
Finding Complex Inner Products
Finding Complex Inner Products
In Exercises 49 and
50, use the inner product
具u, v典 ⴝ u11v11 u12v12 u21v21 u22v22
where
u
u12
v
v12
and v ⴝ 11
u ⴝ 11
u21 u22
v21 v22
to find 具u, v典.
[
]
[
]
冤01 2ii冥 v 冤10 1 2ii冥
1
2i
i 2i
50. u 冤
v冤
冥
1i
0
3i
1冥
49. u 9781133110873_0804.qxp
3/10/12
6:56 AM
Page 419
8.4
51. Let u 共a1 b1i, a2 b2i, . . . , an bn i兲.
(a) Use the definitions of Euclidean norm and Euclidean
inner product to show that
储u储 共 u1 2 u2 2 . . . un 2兲1兾2.
(b) Use the results of part (a) to show that
d共u, v兲 共 u1 v1 2 u2 v2 2
. . . un vn 2兲1兾2.
ⱍ ⱍ
ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
52.
The complex Euclidean inner
product of u and v is sometimes called the
complex dot product. Compare the properties of
the complex dot product in C n and those of the dot
product in R n.
(a) Which properties are the same? Which properties
are different?
(b) Explain the reasons for the differences.
53. Let v1 共i, 0, 0兲 and v2 共i, i, 0兲. If v3 共z1, z2, z3兲
and the set 再v1, v2, v3冎 is not a basis for C 3, what does
this imply about z1, z2, and z3?
54. Let v1 共i, i, i兲 and v2 共1, 0, 1兲. Determine a vector
v3 such that 再v1, v2, v3冎 is a basis for C 3.
In Exercises
55–58, verify the statement using the properties of a
complex inner product.
55. 具u, kv w典 k具u, v典 具u, w典
56. 具u, 0典 0
57. 具u, v典 具u, v典 具v, 2u典
Properties of Complex Inner Products
58. 具u, kv典 k具u, v典
In Exercises 59–63, prove the property, where
u, v, and w are vectors in C n and k is a complex number.
59. 共u v兲 w u w v w
60. 共ku兲 v k共u v兲
61. u 共kv兲 k 共u v兲
62. u u 0
63. u u 0 if and only if u 0.
Proof
64. Writing Let 具u, v典 be a complex inner product and let
k be a complex number. How are 具u, v典 and 具u, kv典
related?
Finding a Linear Transformation In Exercises 65 and
66, determine the linear transformation T : C m → C n that
has the given characteristics.
65. T共1, 0兲 共2 i, 1兲
T共0, 1兲 共0, i兲
66. T共i, 0兲 共2 i, 1兲
T共0, i兲 共0, i兲
Exercises
419
In Exercises 67–70,
the linear transformation T : C m → C n is shown by
T冇v冈 ⴝ Av. Find the image of v and the preimage of w.
1 0
1i
0
67. A , v
, w
i i
1i
0
Finding an Image and a Preimage
冤
68. A 冤
冥
0
i
i
0
冤
冥
1
, v
0
冥
冤冥
冤 冥
i
1
0 , w
1
1i
冤冥
冤 冥 冤 冥 冤冥
冤
冥 冤冥 冤 冥
1
69. A i
i
0
70. A i
0
0
2
2i
, w 2i
0 , v
3 2i
i
3i
1
i
i
1
2
1i
1 , v 5 , w 1 i
0
0
i
71. Find the kernel of the linear transformation in Exercise 68.
72. Find the kernel of the linear transformation in Exercise 69.
In Exercises 73 and 74, find the
image of v ⴝ 冇i, i冈 for the indicated composition, where
T1 and T2 are the matrices below.
0 i
ⴚi
i
and T2 ⴝ
T1 ⴝ
i 0
i ⴚi
Finding an Image
[ ]
73. T2 T1
[
]
74. T1 T2
75. Determine which of the sets below are subspaces of the
vector space of 2 2 complex matrices.
(a) The set of 2 2 symmetric matrices.
(b) The set of 2 2 matrices A satisfying 共 A 兲T A.
(c) The set of 2 2 matrices in which all entries are
real.
(d) The set of 2 2 diagonal matrices.
76. Determine which of the sets below are subspaces of
the vector space of complex-valued functions (see
Example 4).
(a) The set of all functions f satisfying f 共i兲 0.
(b) The set of all functions f satisfying f 共0兲 1.
(c) The set of all functions f satisfying f 共i兲 f 共i兲.
True or False? In Exercises 77 and 78, determine
whether each statement is true or false. If a statement is
true, give a reason or cite an appropriate statement from
the text. If a statement is false, provide an example that
shows the statement is not true in all cases or cite an
appropriate statement from the text.
77. Using the Euclidean inner product of u and v in C n,
u v u1v1 u2v2 . . . unvn.
78. The Euclidean norm of u in C n denoted by 储u储 is
共u u兲2.
9781133110873_0805.qxp
420
3/10/12
Chapter 8
6:55 AM
Page 420
Complex Vector Spaces
8.5 Unitary and Hermitian Matrices
Find the conjugate transpose A* of a complex matrix A.
Determine if a matrix A is unitary.
Find the eigenvalues and eigenvectors of a Hermitian matrix, and
diagonalize a Hermitian matrix.
CONJUGATE TRANSPOSE OF A MATRIX
Problems involving diagonalization of complex matrices and the associated eigenvalue
problems require the concepts of unitary and Hermitian matrices. These matrices
roughly correspond to orthogonal and symmetric real matrices. In order to define
unitary and Hermitian matrices, the concept of the conjugate transpose of a complex
matrix must first be introduced.
Definition of the Conjugate Transpose of a Complex Matrix
The conjugate transpose of a complex matrix A, denoted by A*, is given by
A* A T
where the entries of A are the complex conjugates of the corresponding entries
of A.
Note that if A is a matrix with real entries, then A* AT. To find the conjugate
transpose of a matrix, first calculate the complex conjugate of each entry and then take
the transpose of the matrix, as shown in the following example.
Finding the Conjugate Transpose
of a Complex Matrix
Determine A* for the matrix
A
冤
3 7i
2i
冥
0
.
4i
SOLUTION
A
冤
3 7i
2i
A* AT 冥
0
3 7i
4i
2i
冤3 7i0
冤
冥
0
4i
2i
4i
冥
Several properties of the conjugate transpose of a matrix are listed in the following
theorem. The proofs of these properties are straightforward and are left for you to
supply in Exercises 47–50.
THEOREM 8.8
Properties of the Conjugate Transpose
If A and B are complex matrices and k is a complex number, then the following
properties are true.
1. 共A*兲* A
2. 共A B兲* A* B*
3. 共kA兲* kA*
4. 共AB兲* B*A*
9781133110873_0805.qxp
3/10/12
6:55 AM
Page 421
8.5
Unitary and Hermitian Matrices
421
UNITARY MATRICES
Recall that a real matrix A is orthogonal if and only if A1 AT. In the complex
system, matrices having the property that A1 A* are more useful, and such matrices
are called unitary.
Definition of Unitary Matrix
A complex matrix A is unitary when
A1 A*.
A Unitary Matrix
Show that the matrix A is unitary.
A
1 1i
2 1i
冤
1i
1i
冥
SOLUTION
Begin by finding the product AA*.
冤11 ii
冤40 04冥
1 0
冤
0 1冥
1
2
1
4
AA* 1i 1 1i
1i 2 1i
冥冤
1i
1i
冥
Because
AA* 冤10 01冥 I
2
it follows that A* A1. So, A is a unitary matrix.
Recall from Section 7.3 that a real matrix is orthogonal if and only if its row
(or column) vectors form an orthonormal set. For complex matrices, this property
characterizes matrices that are unitary. Note that a set of vectors
再v1, v2, . . . , vm冎
in C n (a complex Euclidean space) is called orthonormal when the statements below
are true.
1. 储vi储 1, i 1, 2, . . . , m
2. vi vj 0, i j
The proof of the next theorem is similar to the proof of Theorem 7.8 presented in
Section 7.3.
THEOREM 8.9
Unitary Matrices
An n n complex matrix A is unitary if and only if its row (or column) vectors
form an orthonormal set in C n.
9781133110873_0805.qxp
422
3/10/12
Chapter 8
6:55 AM
Page 422
Complex Vector Spaces
The Row Vectors of a Unitary Matrix
Show that the complex matrix A is unitary by showing that its set of row vectors forms
an orthonormal set in C 3.
冤
1i
2
i
冪3
3i
2冪15
1
2
A i
冪3
5i
2冪15
1
2
1
冪3
4 3i
2冪15
冥
SOLUTION
Let r1, r2, and r3 be defined as follows.
1 1i 1
i
i
1
, , r2 ,
,
,
冪
冪
冪
2
2
3
3
3
r1 冢 2,
冣
r3 冢2冪15, 2冪15,
冢
3 i 4 3i
2冪15
5i
冣
冣
Begin by showing that r1, r2 , and r3 are unit vectors.
储r1储 冤 冢12冣冢12冣 冢1 2 i冣冢1 2 i冣 冢 12冣冢 21冣冥
1兾2
冤冢 冪3冣冢 冪3冣 冢冪3冣冢冪3冣 冢 冣冢 冣冥
1 1 1
冤 冥
3 3 3
储r2 储 i
i
i
i
1
冪3
1
冪3
冤 14 42 14冥
1兾2
1兾2
1兾2
1
冤冢2冪5i15冣冢2冪5i15冣 冢23冪15i 冣冢23冪15i 冣 冢42冪153i冣冢42冪153i冣冥
25 10 25
冤 60 60 60冥
储r3储 1兾2
1
Then show that all pairs of distinct vectors are orthogonal.
REMARK
Try showing that the column
vectors of A also form an
orthonormal set in C 3.
1i
i
1
2
冪3
2
i
1
1
i
2冪3 2冪3 2冪3 2冪3
0
r1
r2 冢 2 冣冢 冪3 冣 冢
r1
r3 冢 2 冣冢
r2
r3 冢 冪
1
i
冣冢 冣 冢 冣冢冪13冣
5i
1i 3i
1
2
2
2冪15
2冪15
5i
4 2i 4 3i
4冪15
4冪15
4冪15
0
冣 冢
1
i
冣冢
冣 冢 冣冢42冪153i冣
5i
i
3i
1
4 3i
冢
冢
冣冢
冣
冣冢
冣
冣冢
冪3 2冪15
冪3 2冪15 冣
3 2冪15
5
1 3i 4 3i
6冪5
6冪5
6冪5
0
So, 再r1, r2, r3冎 is an orthonormal set.
1兾2
1
9781133110873_0805.qxp
3/10/12
6:55 AM
Page 423
8.5
Unitary and Hermitian Matrices
423
HERMITIAN MATRICES
A real matrix is symmetric when it is equal to its own transpose. In the complex
system, the more useful type of matrix is one that is equal to its own conjugate
transpose. Such a matrix is called Hermitian after the French mathematician Charles
Hermite (1822–1901).
Definition of a Hermitian Matrix
A square matrix A is Hermitian when
A A*.
As with symmetric matrices, you can recognize Hermitian matrices by inspection.
To see this, consider the 2 2 matrix A.
A
a2i b1 b2i
d1 d2i
1 c2i
冤ac
冥
1
The conjugate transpose of A has the form
A* A T
冤
a1 a2i
冥
c1 c2i
b1 b2i d1 d2i
a a2i c1 c2i
1
.
b1 b2i d1 d2i
冤
冥
If A is Hermitian, then A A*. So, A must be of the form
A
冤b
1
a1 b1 b2i
.
b2i
d1
冥
Similar results can be obtained for Hermitian matrices of order n n. In other words,
a square matrix A is Hermitian if and only if the following two conditions are met.
1. The entries on the main diagonal of A are real.
2. The entry aij in the ith row and the jth column is the complex conjugate of the
entry aji in the jth row and the ith column.
Hermitian Matrices
Which matrices are Hermitian?
1 3i
a.
3i
i
冤
冤
冥
3 2i
3i
c. 2 i
0 1i
3i 1 i
0
冥
b.
冤3 2i0
d.
冤
1
2
3
2
0
1
3 2i
4
冥
3
1
4
冥
SOLUTION
a. This matrix is not Hermitian because it has an imaginary entry on its main diagonal.
b. This matrix is symmetric but not Hermitian because the entry in the first row and
second column is not the complex conjugate of the entry in the second row and first
column.
c. This matrix is Hermitian.
d. This matrix is Hermitian because all real symmetric matrices are Hermitian.
9781133110873_0805.qxp
424
3/10/12
Chapter 8
6:55 AM
Page 424
Complex Vector Spaces
REMARK
Note that this theorem implies
that the eigenvalues of a real
symmetric matrix are real, as
stated in Theorem 7.7.
One of the most important characteristics of Hermitian matrices is that their
eigenvalues are real. This is formally stated in the next theorem.
THEOREM 8.10
The Eigenvalues of a Hermitian Matrix
If A is a Hermitian matrix, then its eigenvalues are real numbers.
PROOF
Let be an eigenvalue of A and let
a1 b1i
a bi
v 2 . 2
..
an bni
冤 冥
be its corresponding eigenvector. If both sides of the equation Av v are multiplied
by the row vector v*, then
v*Av v*共v兲 共v*v兲 共a12 b12 a22 b22 . . . an2 bn2兲.
Furthermore, because
共v*Av兲* v*A*共v*兲* v*Av
it follows that v*Av is a Hermitian 1
number, so is real.
1 matrix. This implies that v*Av is a real
To find the eigenvalues of complex matrices, follow the same procedure as for real
matrices.
Finding the Eigenvalues of a Hermitian Matrix
Find the eigenvalues of the matrix A.
冤
3 2i
3i
A 2i
0 1i
3i 1 i
0
冥
SOLUTION
The characteristic polynomial of A is
ⱍ
3 2 i
3i
2
i
1
i
I
A
ⱍ
ⱍ
3i 1 i
共 3兲共
ⱍ
2兲 共2 i兲关共2 i兲 共3i 3兲兴
3i 关共1 3i兲 3i兴
3
2
共 3 2 6兲 共5 9 3i兲 共3i 9 9兲
3 32 16 12
共 1兲共 6兲共 2兲.
2
So, the characteristic equation is 共 1兲共 6兲共 2兲 0, and the eigenvalues of A
are 1, 6, and 2.
9781133110873_0805.qxp
3/10/12
6:55 AM
Page 425
8.5
Unitary and Hermitian Matrices
425
To find the eigenvectors of a complex matrix, use a procedure similar to that used
for a real matrix. For instance, in Example 5, to find the eigenvector corresponding to
the eigenvalue 1, substitute the value for into the equation
冤
冥冤 冥 冤 冥
3 2 i
3i v1
0
2 i
1 i v2 0
3i 1 i
v3
0
to obtain
冤
TECHNOLOGY
Some graphing utilities and
software programs have built-in
programs for finding the
eigenvalues and corresponding
eigenvectors of complex
matrices.
冥冤 冥 冤 冥
4 2 i
3i v1
0
2 i
1 1 i v2 0 .
3i 1 i
1 v3
0
Solve this equation using Gauss-Jordan elimination, or a graphing utility or software
program, to obtain the eigenvector corresponding to 1 1, which is shown below.
1
v1 1 2i
1
冤 冥
Eigenvectors for 2 6 and 3 2 can be found in a similar manner. They are
冤
1 21i
6 9i
13
冥
LINEAR
ALGEBRA
APPLIED
and
冤
1 3i
2 i , respectively.
5
冥
Quantum mechanics had its start in the early 20th century
as scientists began to study subatomic particles and light.
Collecting data on energy levels of atoms, and the rates of
transition between levels, they found that atoms could be
induced to more excited states by the absorption of light.
German physicist Werner Heisenburg (1901–1976) laid a
mathematical foundation for quantum mechanics using
matrices. Studying the dispersion of light, he used vectors to
represent energy levels of states and Hermitian matrices to
represent “observables” such as momentum, position, and
acceleration. He noticed that a measurement yields precisely
one real value and leaves the system in precisely one of a set
of mutually exclusive (orthogonal) states. So, the eigenvalues
are the possible values that can result from a measurement
of an observable, and the eigenvectors are the corresponding
states of the system following the measurement.
Let matrix A be a diagonal Hermitian matrix that represents
an observable. Then consider a physical system whose
state is represented by the column vector u. To measure the
value of the observable A in the system of state u, you can
find the product
u*Au 关u1
u2
u3兴
冤
a11
0
0
0
a22
0
0
0
a33
冥冤 冥
u1
u2
u3
共u1 u1兲a11 共u2 u2兲a22 共u3u3兲a33.
Because A is Hermitian and its values along the diagonal
are real, u*Au is a real number. It represents the average
of the values given by measuring the observable A on a
system in the state u a large number of times.
Jezper/Shutterstock.com
9781133110873_0805.qxp
426
3/10/12
Chapter 8
6:55 AM
Page 426
Complex Vector Spaces
Just as real symmetric matrices are orthogonally diagonalizable, Hermitian
matrices are unitarily diagonalizable. A square matrix A is unitarily diagonalizable
when there exists a unitary matrix P such that
P1AP
is a diagonal matrix. Because P is unitary, P1 P*, so an equivalent statement is
that A is unitarily diagonalizable when there exists a unitary matrix P such that P*AP
is a diagonal matrix. The next theorem states that Hermitian matrices are unitarily
diagonalizable.
THEOREM 8.11
Hermitian Matrices and Diagonalization
If A is an n n Hermitian matrix, then
1. eigenvectors corresponding to distinct eigenvalues are orthogonal.
2. A is unitarily diagonalizable.
PROOF
To prove part 1, let v1 and v2 be two eigenvectors corresponding to the distinct (and
real) eigenvalues 1 and 2. Because Av1 1v1 and Av2 2v2, you have the
equations shown below for the matrix product 共Av1兲*v2.
共Av1兲*v2 v1*A*v2 v1*Av2 v1*2v2 2v1*v2
共Av1兲*v2 共1v1兲*v2 v1*1v2 1v1*v2
So,
2v1*v2 1v1*v2 0
共 2 1兲v1*v2 0
v1*v2 0
because 1 2
and this shows that v1 and v2 are orthogonal. Part 2 of Theorem 8.11 is often called the
Spectral Theorem, and its proof is left to you.
The Eigenvectors of a Hermitian Matrix
The eigenvectors of the Hermitian matrix shown in Example 5 are mutually orthogonal
because the eigenvalues are distinct. Verify this by calculating the Euclidean inner
products v1 v2, v1 v3, and v2 v3. For example,
v1
v2 共1兲共1 21i兲 共1 2i兲共6 9i兲 共1兲共13兲
共1兲共1 21i兲 共1 2i兲共6 9i兲 13
1 21i 6 9i 12i 18 13
0.
The other two inner products v1
similar manner.
v3
and v2
v3
can be shown to equal zero in a
The three eigenvectors in Example 6 are mutually orthogonal because they
correspond to distinct eigenvalues of the Hermitian matrix A. Two or more eigenvectors
corresponding to the same eigenvalue may not be orthogonal. Once any set of linearly
independent eigenvectors is obtained for an eigenvalue, however, the Gram-Schmidt
orthonormalization process can be used to find an orthogonal set.
9781133110873_0805.qxp
3/10/12
6:55 AM
Page 427
8.5
Unitary and Hermitian Matrices
427
Diagonalization of a Hermitian Matrix
Find a unitary matrix P such that P*AP is a diagonal matrix where
冤
3
A 2i
3i
冥
2i
0
1i
3i
1i .
0
SOLUTION
The eigenvectors of A are shown after Example 5. Form the matrix P by normalizing
these three eigenvectors and using the results to create the columns of P.
储v1储 储共1, 1 2i, 1兲储 冪1 5 1 冪7
储v2储 储共1 21i, 6 9i, 13兲储 冪442 117 169 冪728
储v3储 储共1 3i, 2 i, 5兲储 冪10 5 25 冪40
So,
P
冤
1
冪7
1 2i
冪7
1
冪7
冥
1 21i
冪728
6 9i
冪728
13
冪728
1 3i
冪40
2 i
.
冪40
5
冪40
Try computing the product P*AP for the matrices A and P in Example 7 to see that
P*AP 冤
1
0
0
0
6
0
0
0
2
冥
where 1, 6, and 2 are the eigenvalues of A.
You have seen that Hermitian matrices are unitarily diagonalizable. It turns out
that there is a larger class of matrices, called normal matrices, that are also unitarily
diagonalizable. A square complex matrix A is normal when it commutes with its
conjugate transpose: AA* A*A. The main theorem of normal matrices states that
a complex matrix A is normal if and only if it is unitarily diagonalizable. You are asked
to explore normal matrices further in Exercise 56.
The properties of complex matrices described in this section are comparable to the
properties of real matrices discussed in Chapter 7. The summary below indicates the
correspondence between unitary and Hermitian complex matrices when compared with
orthogonal and symmetric real matrices.
Comparison of Symmetric and Hermitian Matrices
A is a symmetric matrix
(real)
1. Eigenvalues of A are real.
2. Eigenvectors corresponding
to distinct eigenvalues are
orthogonal.
3. There exists an orthogonal
matrix P such that
PTAP
is diagonal.
A is a Hermitian matrix
(complex)
1. Eigenvalues of A are real.
2. Eigenvectors corresponding
to distinct eigenvalues are
orthogonal.
3. There exists a unitary matrix P
such that
P*AP
is diagonal.
9781133110873_0805.qxp
428
3/10/12
Chapter 8
6:55 AM
Page 428
Complex Vector Spaces
8.5 Exercises
In Exercises 1–4,
determine the conjugate transpose of the matrix.
Finding the Conjugate Transpose
1.
冤2
i
i
3i
冥
2.
冤
0 5 i 冪2i
3.
5i
6
4
冪
2i
4
3
冥
冤1 2i1
冤
2
4. 5
0
2i
1
冥
i
3i
6i
16. A 冥
In Exercises 5 and
6, use a software program or graphing utility to find the
conjugate transpose of the matrix.
Finding the Conjugate Transpose
冤
冤
1i
2i
5.
1i
i
0
1
i
2i
2i
0
6.
i
1 2i
1
0
2
1
1
2i
i
0
1
2i
2i
4
i
2i
4i
0
冥
2i
1i
1
2i
冤0
i
0
0
冥
8. A 20. A 冤
1i 0
9. A 冪2
0 1
冤
0
i
冪3
冪2
1
2
i
1
冥
13. A 15. A 冤11 ii
冤i0
1i
1i
冥
冥
冥
0
i
冤 冥
4
5
3
i
5
3
5
4
i
5
12. A 冤11 ii
14. A 冤
1i
1i
冥
冥
i
i
冪2
冪2
i
冪2
i
i
冪2
冪3
冪6
i
0
冪3
i
冪6
冥
1
冤
冥
冪3 i
冥
1 冪3i
1 冪3i
2冪2 冪3 i
冤
冤
1 i 1 i
2
2
18. A 1
1
冪2
冪2
3
i
5
4
i
5
0
1 i
冪6
2
冪6
1
0
1i
冪3
1
冪3
0
0
i
冪2
冤i0 0i 冥
22.
0
23. 2 i
1
冤
2i
i
0
1
0
1
冤2 1i
2i
2
3i
3i
25.
Identifying Unitary Matrices In Exercises 11–16,
determine whether A is unitary by calculating AA*.
11. A i
冥
Identifying Hermitian Matrices In Exercises 21–26,
determine whether the matrix is Hermitian.
冥
1
21.
1
1i
2
2
1
i
冪3
冪3
1
1i
2
2
1
2
10. A i
冤i
i
冪6
冤 冥
the matrix is not unitary.
7. A i
冪3
4
5
17. A 3
5
In Exercises 7–10, explain why
Non-Unitary Matrices
i
冪2
Row Vectors of a Unitary Matrix In Exercises 17–20,
(a) verify that A is unitary by showing that its rows are
orthonormal, and (b) determine the inverse of A.
19. A 冥
冤
冤
冥
冤
冥
0
i
0
24. 2 i
0
i
i
1
1
0
0
冥
冥
冪2 i
1
26. 冪2 i
5
冤0i
5
3i
6
2
3i
冥
Finding Eigenvalues of a Hermitian Matrix In
Exercises 27–32, determine the eigenvalues of the matrix A.
27. A 冤i0 0i冥
29. A 冤1 3i
冤
1
31. A 0
0
4
i
0
1i
2
冥
1i
3i
2i
冥
28. A 冤i3 3i冥
30. A 冤2 0i
2i
4
冥
9781133110873_0805.qxp
3/10/12
6:55 AM
Page 429
8.5
32. A 冤
2
i
冪2
冪2
2
0
0
2
i
冪2
冥
i
i
冪2
Unitary Matrices In Exercises 45 and 46, use the result of
Exercise 44 to determine a, b, and c such that A is unitary.
1 1
45. A b
冪2
冤
33. Exercise 27
34. Exercise 30
35. Exercise 31
36. Exercise 28
In Exercises
37–41, find a unitary matrix P that diagonalizes the
matrix A.
Diagonalization of a Hermitian Matrix
冤i0 0i冥
39. A 冤
2
38. A i
冪2
冪2
2
0
0
2
i
冪2
i
冪2
冤2 0i
2i
4
冥
冥
i
4 2 2i
40. A 2 2i
6
冤
41. A 冤
冥
1
0
0
0
0
1 1 i
1 i
0
42.
冥
Consider the following matrix.
冤
2 3 i 4 i
A 3i
1 1i
4i 1i
3
Is A unitary? Explain.
Is A Hermitian? Explain.
Are the row vectors of A orthonormal? Explain.
The eigenvalues of A are distinct. Is it possible
to determine the inner products of the pairs of
eigenvectors by inspection? If so, state the
value(s). If not, explain why not.
(e) Is A unitarily diagonalizable? Explain.
43. Show that A In is unitary by computing AA*.
44. Let z be a complex number with modulus 1. Show that
the matrix A is unitary.
冤
冥
冥
51. Proof Let A be a matrix such that A* A O. Prove
that iA is Hermitian.
52. Show that det共A兲 det共A兲, where A is a 2 2 matrix.
Determinants In Exercises 53 and 54, assume that the
result of Exercise 52 is true for matrices of any size.
53. Show that det共A*兲 det共A兲.
54. Prove that if A is unitary, then det共A兲 1.
ⱍ
ⱍ
55. (a) Prove that every Hermitian matrix A can be written as
the sum A B iC, where B is a real symmetric
matrix and C is real and skew-symmetric.
(b) Use part (a) to write the matrix
A
冤1 2i
1i
3
冥
as the sum A B iC, where B is a real symmetric
matrix and C is real and skew-symmetric.
(c) Prove that every n n complex matrix A can be
written as A B iC, where B and C are
Hermitian.
(d) Use part (c) to write the complex matrix
A
56. (a)
(b)
(c)
(d)
(e)
(f)
1 z
z
冪2 iz iz
冥
6 3i
冪45
c
In Exercises 47–50, prove the formula, where A
and B are n ⴛ n complex matrices.
47. 共A*兲* A
48. 共A B兲* A* B*
49. 共kA兲* kA*
50. 共AB兲* B*A*
冥
(a)
(b)
(c)
(d)
A
冤
1 a
46. A 冪2
b
a
c
Proof
In
Exercises 33 – 36, determine the eigenvectors of the
matrix in the indicated exercise.
Finding Eigenvectors of a Hermitian Matrix
37. A 429
Exercises
(g)
冤2 ii
冥
2
1 2i
as the sum A B iC, where B and C are
Hermitian.
Prove that every Hermitian matrix is normal.
Prove that every unitary matrix is normal.
Find a 2 2 matrix that is Hermitian, but not unitary.
Find a 2 2 matrix that is unitary, but not
Hermitian.
Find a 2 2 matrix that is normal, but neither
Hermitian nor unitary.
Find the eigenvalues and corresponding eigenvectors
of your matrix in part (e).
Show that the complex matrix
冤0i 1i冥
is not diagonalizable. Is this matrix normal?
9781133110873_08REV.qxp
430
3/10/12
Chapter 8
8
6:53 AM
Page 430
Complex Vector Spaces
Review Exercises
Operations with Complex Numbers
In Exercises
1–6, perform the operation.
1. Find u z: u 2 4i, z 4i
2. Find u z: u 4, z 8i
3. Find uz: u 4 2i, z 4 2i
4. Find uz: u 2i, z 1 2i
u
5. Find : u 6 2i, z 3 3i
z
u
6. Find : u 7 i, z i
z
Converting to Standard Form In Exercises 37–42,
find the standard form of the complex number.
In Exercises
11–18, perform the operation using
4ⴚi
2
1ⴙi
i
Aⴝ
and B ⴝ
.
3
3ⴙi
2i 2 ⴙ i
11. A B
12. A B
13. 2iB
14. iA
15. det共A B兲
16. det共A B兲
17. 3BA
18. 2AB
冤
冥
冤
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
In Exercises 25 –28,
perform the indicated operation.
2i
2i
27.
共1 2i兲共1 2i兲
3 3i
1i
1 2i
28.
5 2i
共2 2i兲共2 3i兲
29 and 30, find Aⴚ1 (if it exists).
3i
1 2i
29. A 23
11
5 5i
2 3i
冤
30. A 冥
冤0
5
1i
i
冥
5
5
i sin
4
4
冢
3
3
i sin
2
2
41. 7 cos
冢
2
2
i sin
3
3
冣
40. 6 cos
冣
42. 4关cos i sin 兴
冣
In Exercises
43–46, perform the indicated operation. Leave the result
in polar form.
Multiplying and Dividing in Polar Form
i sin
2
2
冣冥冤3冢cos 6 i sin 6 冣冥
44.
冤 2 冢cos 2 i sin 2 冣冥冤2 冢cos 冢 2 冣 i sin 冢 2 冣冣冥
45.
9 关cos共兾2兲 i sin 共兾2兲兴
6 关cos共2兾3兲 i sin 共2兾3兲兴
46.
4 关cos 共兾4兲 i sin 共兾4兲兴
7 关cos 共兾3兲 i sin 共兾3兲兴
1
In Exercises
47–50, find the indicated power of the number and
express the result in polar form.
47. 共1 i兲4
48. 共2i兲3
Finding Powers of Complex Numbers
冤 冢
49. 冪2 cos
26.
Finding the Inverse of a Complex Matrix
冢
39. 4 cos
43. 4 cos
In Exercises
19 – 24, perform the operation using w ⴝ 2 ⴚ 2i,
v ⴝ 3 ⴙ i, and z ⴝ ⴚ1 ⴙ 2i.
19. z
20. v
21. w
22. vz
23. wv
24. zw
25.
冤 冢 3 冣 i sin冢 3 冣冥
38. 2 cos 冤冢
冥
Operations with Conjugates and Moduli
Dividing Complex Numbers
冤 冢 6 冣 i sin冢 6 冣冥
37. 5 cos Finding Zeros In Exercises 7–10, use the given zero to
find all zeros of the polynomial function.
7. p共x兲 x 3 2x 2 2x 4
Zero: 冪2i
8. p共x兲 x3 2x 4
Zero: 1 i
4
3
2
9. p共x兲 x x 3x 5x 10
Zero: 冪5i
4
3
2
10. p共x兲 x x x 3x 6
Zero: 冪3i
Operations with Complex Matrices
In Exercises 31– 36,
determine the polar form of the complex number.
31. 4 4i
32. 2 2i
33. 冪3 i
34. 1 冪3i
35. 7 4i
36. 3 2i
Converting to Polar Form
i sin
6
6
冣冥
7
冤冢
50. 5 cos
i sin
3
3
Finding Roots of Complex Numbers
51–54, express the roots in standard form.
冢
51. Square roots: 25 cos
In Exercises
冢
52. Cube roots: 27 cos
2
2
i sin
3
3
i sin
6
6
冣
53. Cube roots: i
冢
54. Fourth roots: 16 cos
i sin
4
4
冣
冣
冣冥
4
In Exercises
9781133110873_08REV.qxp
3/10/12
6:53 AM
Page 431
Review Exercises
Vector Operations in C n
In Exercises 55–58, find the
indicated vector using u ⴝ 冇4i, 2 ⴙ i冈, v ⴝ 冇3, ⴚi 冈, and
w ⴝ 冇3 ⴚ i, 4 ⴙ i冈.
55. 7u v
56. 3iw 共4 i兲v
57. iu iv iw
58. 共3 2i兲u 共2i兲w
Finding the Euclidean Norm In Exercises 59 and 60,
determine the Euclidean norm of the vector.
59. v 共3 5i, 2i兲
60. v 共3i, 1 5i, 3 2i兲
In Exercises 61 and
62, find the Euclidean distance between the vectors.
61. v 共2 i, i兲, u 共i, 2 i兲
62. v 共2 i, 1 2i, 3i兲, u 共4 2i, 3 2i, 4兲
冤
9
76. 2 i
2
2i
0
1 i
Finding the Conjugate Transpose In Exercises 67–70,
determine the conjugate transpose of the matrix.
1 4i 3 i
67. A 3i 2i
冤
2i
68. A 冤
1 2i
冤
冤
冥
2i
2 2i 冥
2i
3 2i
2i
5
69. A 2 2i
3i
2
70. A i
1
1i
2 2i
1i
i
0
2i
3 2i
i
1 2i
冥
冥
Identifying Unitary Matrices In Exercises 71–74,
determine whether the matrix is unitary.
i 1
2i 1i
4
4
冪2
冪2
71.
72.
冪2
i
1
i
冪2
冪2
冪3
冪3
1
1
0
冪2
冪2
1
0
73.
74.
0 i
0
i i
1i
1 i
0
2
2
冤
冤
冤
冥
冤
冥
冥
冥
In Exercises 75 and
76, determine whether the matrix is Hermitian.
1
1 i 2 i
75. 1 i
3
i
2i
i
4
Identifying Hermitian Matrices
冤
冥
冥
In Exercises
77 and 78, find the eigenvalues and corresponding
eigenvectors of the matrix.
Finding Eigenvalues and Eigenvectors
77.
冤
4
2i
2i
0
冥
Finding the Euclidean Distance
Finding Complex Inner Products In Exercises 63–66,
use the inner product 具u, v典 ⴝ u1v1 ⴙ 2u2v2 to find 具u, v典.
63. u 共i, 3i兲 and v 共2i, 2i兲
64. u 共i, 2 i兲 and v 共5 i, i兲
65. u 共1 i, 1 2i兲 and v 共2 i, 1 2i兲
66. u 共2 2i, 1兲 and v 共3 4i, 2兲
2
1 i
3
431
冤
2 0
78. 0 3
i 0
i
0
2
冥
79. Proof Prove that if A is an invertible matrix, then A*
is also invertible.
80. Determine all complex numbers z such that z z.
81. Proof Prove that if z is a zero of a polynomial
equation with real coefficients, then the conjugate of z
is also a zero.
82. (a) Find the determinant of the Hermitian matrix
冤
冥
2i
3i
0
1i .
1i
0
(b) Proof Prove that the determinant of any
Hermitian matrix is real.
83. Proof Let A and B be Hermitian matrices. Prove that
AB BA if and only if AB is Hermitian.
84. Proof Let u be a unit vector in C n. Define
H I 2uu*. Prove that H is an n n Hermitian and
unitary matrix.
85. Proof Use mathematical induction to prove
DeMoivre’s Theorem.
86. Proof Show that if z1 z2 and z1z2 are both nonzero
real numbers, then z1 and z2 are both real numbers.
87. Proof Prove that if z and w are complex numbers,
then
zw z w.
88. Proof Prove that for all vectors u and v in a complex
inner product space,
1
具u, v典 4 关 储u v储2 储u v储2 i储u iv储2
i储u iv储2兴.
ⱍ
3
2i
3i
ⱍ ⱍⱍ ⱍ ⱍ
True or False? In Exercises 89 and 90, determine
whether each statement is true or false. If a statement is
true, give a reason or cite an appropriate statement from
the text. If a statement is false, provide an example that
shows the statement is not true in all cases or cite an
appropriate statement from the text.
89. A square complex matrix A is called normal if it
commutes with its conjugate transpose so that
AA* A*A.
90. A square complex matrix A is called Hermitian if
A A*.
9781133110873_08REV.qxp
432
8
3/10/12
Chapter 8
6:53 AM
Page 432
Complex Vector Spaces
Project
1 Population Growth and Dynamical Systems (II)
In the projects for Chapter 7, you were asked to model the populations of two
species using a system of differential equations of the form
y 1
共t兲 ay1共t兲 by2共t兲
y 2
共t兲 cy1共t兲 dy2共t兲.
The constants a, b, c, and d depend on the particular species being studied. In
Chapter 7, you looked at an example of a predator-prey relationship, in which
a 0.5, b 0.6, c 0.4, and d 3.0. Now consider a slightly different model.
y 1
共t兲 0.6y1共t兲 0.8y2共t兲,
y1共0兲 36
y 2
共t兲 0.8y1共t兲 0.6y2共t兲,
y2共0兲 121
1. Use the diagonalization technique to find the general solutions y1共t兲 and y2共t兲 at
any time t > 0. Although the eigenvalues and eigenvectors of the matrix
A
0.6
冤0.8
冥
0.8
0.6
are complex, the same principles apply, and you can obtain complex exponential
solutions.
2. Convert the complex solutions to real solutions by observing that if a bi
is a (complex) eigenvalue of A with (complex) eigenvector v, then the real and
imaginary parts of etv form a linearly independent pair of (real) solutions. Use
the formula ei cos i sin .
3. Use the initial conditions to find the explicit form of the (real) solutions of the
original equations.
4. If you have access to a graphing utility or software program, graph the solutions
obtained in part 3 over the domain 0 t 3. At what moment are the two
populations equal?
5. Interpret the solution in terms of the long-term population trend for the two
species. Does one species ultimately disappear? Why or why not? Contrast this
solution to that obtained for the model in Chapter 7.
6. If you have access to a graphing utility or software program that can numerically
solve differential equations, use it to graph the solutions of the original system
of equations. Does this numerical approximation appear to be accurate?
Supri Suharjoto/Shutterstock.com