complex vector spaces - cengage · 482 chapter 8 complex vector spaces complex numbers ... 9 12i 4...

8 Complex Vector Spaces

8.1 Complex Numbers8.2 Conjugates and Division

of Complex Numbers8.3 Polar Form and

DeMoivre’s Theorem8.4 Complex Vector Spaces

and Inner Products8.5 Unitary and Hermitian

Matrices

CHAPTER OBJECTIVES

■ Graphically represent complex numbers in the complex plane.

■ Perform operations with complex numbers.

■ Represent complex numbers as vectors.

■ Use the Quadratic Formula to find all zeros of a quadratic polynomial.

■ Perform operations with complex matrices.

■ Find the determinant of a complex matrix.

■ Find the conjugate, modulus, and argument of a complex number.

■ Multiply and divide complex numbers.

■ Find the inverse of a complex matrix.

■ Determine the polar form of a complex number.

■ Convert a complex number from standard form to polar form and from polar form to standard form.

■ Multiply and divide complex numbers in polar form.

■ Find roots and powers of complex numbers in polar form.

■ Use DeMoivre’s Theorem to find roots of complex numbers in polar form.

■ Recognize complex vector spaces,

■ Perform vector operations in

■ Represent a vector in by a basis.

■ Find the Euclidian inner product and the Euclidian norm of a vector in

■ Find the Euclidian distance between two vectors in

■ Find the conjugate transpose of a complex matrix

■ Determine if a matrix is unitary or Hermitian.

■ Find the eigenvalues and eigenvectors of a Hermitian matrix.

■ Diagonalize a Hermitian matrix.

■ Determine if a Hermitian matrix is normal.

A

A.A*

C n.

C n.

C n

C n.

C n.

481

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482 Chapter 8 Complex Vector Spaces

Complex Numbers

So far in the text, the scalar quantities used have been real numbers. In this chapter, youwill expand the set of scalars to include complex numbers.

In algebra it is often necessary to solve quadratic equations such as The general quadratic equation is and its solutions are given by theQuadratic Formula,

where the quantity under the radical, is called the discriminant. Ifthen the solutions are ordinary real numbers. But what can you conclude

about the solutions of a quadratic equation whose discriminant is negative? For example, theequation has a discriminant of From your experience withordinary algebra, it is clear that there is no real number whose square is By writing

you can see that the essence of the problem is that there is no real number whose square isTo solve the problem, mathematicians invented the imaginary unit which has the

property In terms of this imaginary unit, you can write

The imaginary unit is defined as follows.

R E M A R K : When working with products involving square roots of negative numbers, besure to convert to a multiple of before multiplying. For instance, consider the followingoperations.

Correct

Incorrect

With this single addition of the imaginary unit to the real number system, the systemof complex numbers can be developed.

i

��1��1 � ��1��1� � �1 � 1

��1��1 � i � i � i2 � �1

i

i

��16 � 4��1 � 4i.

i2 � �1.i,�1.

��16 � �16��1� � �16��1 � 4��1,

�16.b2 � 4ac � �16.x2 � 4 � 0

b2 � 4ac � 0,b2 � 4ac,

x ��b � �b2 � 4ac

2a,

ax2 � bx � c � 0,x2 � 3x � 2 � 0.

8.1

The number is called the imaginary unit and is defined as

where i2 � �1.

i � ��1

iDefinition of the Imaginary Unit i

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Sect ion 8 .1 Complex Numbers 483

Some examples of complex numbers written in standard form are and The set of real numbers is a subset of the set of complex numbers. Tosee this, note that every real number can be written as a complex number using That is, for every real number,

A complex number is uniquely determined by its real and imaginary parts. So, you cansay that two complex numbers are equal if and only if their real and imaginary parts areequal. That is, if and are two complex numbers written in standard form,then

if and only if and

The Complex Plane

Because a complex number is uniquely determined by its real and imaginary parts, it is natural to associate the number with the ordered pair With this association,you can graphically represent complex numbers as points in a coordinate plane called thecomplex 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.For instance, Figure 8.1 shows the graph of two complex numbers, and Thenumber is associated with the point and the number is associated withthe point

Another way to represent the complex number is as a vector whose horizontalcomponent is and whose vertical component is (See Figure 8.2.) (Note that the use of the letter to represent the imaginary unit is unrelated to the use of to represent a unit vector.)

iib.aa � bi

��2, �1�.�2 � i�3, 2�3 � 2i

�2 � i.3 � 2i

�a, b�.a � bi

b � d.a � c

a � bi � c � di

c � dia � bi

a � a � 0i.b � 0.a

�6i � 0 � 6i.4 � 3i,2 � 2 � 0i,

If and are real numbers, then the number

is a complex number, where is the real part and is the imaginary part of thenumber. The form is the standard form of a complex number.a � bi

bia

a � bi

baDefinition of a Complex Number

(3, 2) or 3 + 2i

Realaxis

Imaginaryaxis

1−1

1

2

3

2 3−2

(−2, −1)or −2 − iThe Complex Plane

Figure 8.1

y1

x1

1

−1

−2

−3

4 − 2iVertical

component

Vector Representation of aComplex Number

Horizontalcomponent

Figure 8.2

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Addition and Scalar Multiplication of Complex Numbers

Because a complex number consists of a real part added to a multiple of the operationsof addition and multiplication are defined in a manner consistent with the rules for operat-ing with real numbers. For instance, to add (or subtract) two complex numbers, add (or subtract) the real and imaginary parts separately.

(a)

(b)

R E M A R K : Note in part (a) of Example 1 that the sum of two complex numbers can bea real number.

Using the vector representation of complex numbers, you can add or subtract two complex numbers geometrically using the parallelogram rule for vector addition, as shownin Figure 8.3.

Figure 8.3

Realaxis

1

1

2

−3

−3 2 3

Imaginaryaxis

w = 3 + i

z = 1 − 3i

z − w = −2 − 4iSubtraction of Complex Numbers

1

−2

−3

−4

−1 2 3 41 65

2

3

4

Addition of Complex Numbersw = 2 − 4i

z + w = 5

z = 3 + 4i

Imaginaryaxis

Realaxis

�1 � 3i� � �3 � i� � �1 � 3� � ��3 � 1�i � �2 � 4i

�2 � 4i� � �3 � 4i� � �2 � 3� � ��4 � 4�i � 5

E X A M P L E 1 Adding and Subtracting Complex Numbers

i,

The sum and difference of and are defined as follows.

Sum

Difference�a � bi� � �c � di� � �a � c� � �b � d �i

�a � bi� � �c � di� � �a � c� � �b � d �i

c � dia � biDefinition of Addition and Subtraction of Complex Numbers

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Many of the properties of addition of real numbers are valid for complex numbers aswell. 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,

To multiply a complex number by a real scalar, use the definition below.

Geometrically, multiplication of a complex number by a real scalar corresponds to themultiplication of a vector by a scalar, as shown in Figure 8.4.

(a)

(b)

Multiplication of a Complex Number by a Real Number

Figure 8.4

With addition and scalar multiplication, the set of complex numbers forms a vector spaceof dimension 2 (where the scalars are the real numbers). You are asked to verify this inExercise 57.

Imaginaryaxis

Realaxis

1

1

2

−2

−3

3

2 3

−z = −3 − i

z = 3 + i

Realaxis

z = 3 + i

2z = 6 + 2i

1

1

2

−1

−2

3

4

2 3 4 65

Imaginaryaxis

� �1 � 6i �4�1 � i� � 2�3 � i� � 3�1 � 4i� � �4 � 4i � 6 � 2i � 3 � 12i

� 38 � 17i

3�2 � 7i� � 4�8 � i� � 6 � 21i � 32 � 4i

E X A M P L E 2 Operations with Complex Numbers

� 3 � 3i.

�2 � i� � �3 � 2i� � ��2 � 4i� � �2 � 3 � 2� � �1 � 2 � 4�i

If is a real number and is a complex number, then the scalar multiple of andis defined as

c�a � bi� � ca � cbi.

a � bica � bicDefinition of

Scalar Multiplication

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Multiplication of Complex Numbers

The operations of addition, subtraction, and multiplication by a real number have exact counterparts with the corresponding vector operations. By contrast, there is no directcounterpart for the multiplication of two complex numbers.

Rather than try to memorize this definition of the product of two complex numbers, youshould simply apply the distributive property, as follows.

Distributive property


Use

Commutative property


This is demonstrated in the next example.

(a)

(b)

Use the Quadratic Formula to find the zeros of the polynomial

and verify that for each zero.p�x� � 0

p�x� � x2 � 6x � 13

E X A M P L E 4 Complex Zeros of a Polynomial

� 11 � 2i � 8 � 3 � 6i � 4i

� 8 � 6i � 4i � 3��1� �2 � i��4 � 3i� � 8 � 6i � 4i � 3i2

��2��1 � 3i� � �2 � 6i

E X A M P L E 3 Multiplying Complex Numbers

� �ac � bd � � �ad � bc�i � ac � bd � �ad �i � �bc�i

i 2 � �1. � ac � �ad �i � �bc�i � �bd ��1� � ac � �ad �i � �bc�i � �bd �i2

�a � bi��c � di� � a�c � di� � bi�c � di�

Many computer software programs and graphing utilities are capable of calculating with complexnumbers. For example, on some graphing utilities, you can express a complex number as anordered pair Try verifying the result of Example 3(b) by multiplying and Youshould obtain the ordered pair �11, 2�.

�4, 3�.�2, �1��a, b�.a � bi

TechnologyNote

The product of the complex numbers and is defined as

�a � bi��c � di� � �ac � bd � � �ad � bc�i .

c � dia � biDefinition of Multiplication of

Complex Numbers

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S O L U T I O N Using the Quadratic Formula, you have

Substituting these values of into the polynomial , you have

and

In Example 4, the two complex numbers and are complex conjugates ofeach other (together they form a conjugate pair). A well-known result from algebra statesthat the complex zeros of a polynomial with real coefficients must occur in conjugate pairs.(See Review Exercise 86.) More will be said about complex conjugates in Section 8.2.

Complex Matrices

Now that you are able to add, subtract, and multiply complex numbers, you can apply theseoperations to matrices whose entries are complex numbers. Such a matrix is called complex.

All of the ordinary operations with matrices also work with complex matrices, asdemonstrated in the next two examples.

3 � 2i3 � 2i

� 9 � 12i � 4 � 18 � 12i � 13 � 0.

� 9 � 6i � 6i � 4 � 18 � 12i � 13

� �3 � 2i��3 � 2i� � 6�3 � 2i� � 13

p�3 � 2i� � �3 � 2i�2 � 6�3 � 2i� � 13

� 9 � 12i � 4 � 18 � 12i � 13 � 0

� 9 � 6i � 6i � 4 � 18 � 12i � 13

� �3 � 2i��3 � 2i� � 6�3 � 2i� � 13

p�3 � 2i� � �3 � 2i�2 � 6�3 � 2i� � 13

p�x�x

�6 � ��16

2�

6 � 4i

2� 3 � 2i.

x ��b � �b2 � 4ac

2a�

6 � �36 � 52

2

A matrix whose entries are complex numbers is called a complex matrix.Definition of a Complex Matrix

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Let and be the complex matrices below

and

and determine each of the following.

(a) (b) (c) (d)

S O L U T I O N (a)

(b)

(c)

(d)

Find the determinant of the matrix

S O L U T I O N

� �8 � 26i

� 10 � 20i � 6i � 12 � 6

det�A� � �2 � 4i3

25 � 3i� � �2 � 4i��5 � 3i� � �2��3�

A � �2 � 4i3

25 � 3i�.

E X A M P L E 6 Finding the Determinant of a Complex Matrix

� � �2 � 0�1 � 2 � 3i � 4i � 6

2i � 2 � 0i � 1 � 4 � 8i� � � �2

7 � i�2 � 2i

3 � 9i�

BA � �2ii

01 � 2i��

i2 � 3i

1 � i4�

A � B � � i2 � 3i

1 � i4� � �2i

i0

1 � 2i� � � 3i2 � 2i

1 � i5 � 2i�

�2 � i�B � �2 � i��2ii

01 � 2i� � �2 � 4i

1 � 2i0

4 � 3i�

3A � 3� i2 � 3i

1 � i4� � � 3i

6 � 9i3 � 3i

12�BAA � B�2 � i�B3A

B � �2ii

01 � 2i�A � � i

2 � 3i1 � i

4�BA

E X A M P L E 5 Operations with Complex Matrices

Many computer software programs and graphing utilities are capable of performing matrix operations on complex matrices. Try verifying the calculation of the determinant of the matrix inExample 6. You should obtain the same answer, ��8, �26�.

TechnologyNote

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ExercisesSECTION 8.1In Exercises 1–6, determine the value of the expression.

1. 2.

3. 4.

5. 6.

In Exercises 7–12, plot the complex number.

7. 8.

9. 10.

11. 12.

In Exercises 13 and 14, use vectors to illustrate the operationsgraphically. Be sure to graph the original vector.

13. and , where

14. and , where

In Exercises 15–18, determine such that the complex numbers ineach pair are equal.

15.

16.

17.

18.

In Exercises 19–26, find the sum or difference of the complex numbers. Use vectors to illustrate your answer graphically.

19. 20.

21. 22.

23. 24.

25. 26.

In Exercises 27–36, find the product.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

In Exercises 37–42, determine the zeros of the polynomial function.

37.

38.

39.

40.

41.

42.

In Exercises 43–46, use the given zero to find all zeros of the polynomial function.

43. Zero:

44. Zero:

45. Zero:

46. Zero:

In Exercises 47–56, perform the indicated matrix operation usingthe complex matrices and

and

47. 48.

49. 50.

51. 52.

53. det 54. det

55. 56.

57. Prove that the set of complex numbers, with the operations ofaddition and scalar multiplication (with real scalars), is a vectorspace of dimension 2.

58. (a) Evaluate for and 5.(b) Calculate (c) Find a general formula for for any positive integer

59. Let

(a) Calculate for and 5.(b) Calculate (c) Find a general formula for for any positive integer n.An

A2010.n � 1, 2, 3, 4,An

A � �0i

i0�.

n.in

i 2010.n � 1, 2, 3, 4,in

BA5AB

�B��A � B�

14iB2iA

12B2A

B � AA � B

B � �1 � i�3

3i�i�A � � 1 � i

2 � 2i1

�3i�B.A

x � 3ip�x� � x3 � x2 � 9x � 9

x � 5ip�x� � 2x3 � 3x2 � 50x � 75

x � �4p�x� � x3 � 2x2 � 11x � 52

x � 1p�x� � x3 � 3x2 � 4x � 2

p�x� � x4 � 10x 2 � 9

p�x� � x4 � 16

p�x� � x 2 � 4x � 5

p�x� � x 2 � 5x � 6

p�x� � x 2 � x � 1

p�x� � 2x 2 � 2x � 5

�1 � i�2�1 � i�2�a � bi�3

�2 � i��2 � 2i��4 � i��1 � i�3

�a � bi��a � bi��a � bi�2

�4 � �2 i��4 � �2 i ��7 � i��7 � i��3 � i�� 2

3 � i��5 � 5i��1 � 3i�

�2 � i� � �2 � i��2 � i� � �2 � i��12 � 7i� � �3 � 4i�6 � ��2i�i � �3 � i��5 � i� � �5 � i��1 � �2 i� � �2 � �2 i��2 � 6i� � �3 � 3i�

��x � 4� � �x � 1�i, x � 3i

�x 2 � 6� � �2x�i, 15 � 6i

�2x � 8� � �x � 1�i, 2 � 4i

x � 3i, 6 � 3i

x

u � 2 � i�32u3u

u � 3 � i2u�u

z � 1 � 5iz � 1 � 5i

z � 7z � �5 � 5i

z � 3iz � 6 � 2i

��i�7i 4

i 3��4��4

�8��8��2��3

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60. Prove that if the product of two complex numbers is zero, thenat least one of the numbers must be zero.

True or False? In Exercises 61 and 62, determine whether eachstatement is true or false. If a statement is true, give a reason or citean appropriate statement from the text. If a statement is false,provide an example that shows the statement is not true in all casesor cite an appropriate statement from the text.

61.

62. ��10�2 � �100 � 10

��2��2 � �4 � 2

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Sect ion 8 .2 Conjugates and Div is ion of Complex Numbers 491

Conjugates and Division of Complex Numbers

In Section 8.1, it was mentioned that the complex zeros of a polynomial with real coeffi-cients occur in conjugate pairs. For instance, in Example 4 you saw that the zeros of

are and 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.

From this definition, you can see that the conjugate of a complex number is found bychanging the sign of the imaginary part of the number, as demonstrated in the next example.

Complex Number Conjugate

(a)(b)(c)(d)

R E M A R K : In part (d) of Example 1, note that 5 is its own complex conjugate. In gen-eral, it can be shown that a number is its own complex conjugate if and only if the numberis real. (See Exercise 39.)

Geometrically, two points in the complex plane are conjugates if and only if they are reflections about the real (horizontal) axis, as shown in Figure 8.5 on the next page.

z � 5z � 5z � 2iz � �2iz � 4 � 5iz � 4 � 5iz � �2 � 3iz � �2 � 3i

E X A M P L E 1 Finding the Conjugate of a Complex Number

3 � 2i.3 � 2ip�x� � x2 � 6x � 13

8.2

The conjugate of the complex number is denoted by and is given by

z � a � bi.

zz � a � biDefinition of the Conjugate of a

Complex Number

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Conjugate of a Complex Number

Figure 8.5

Complex conjugates have many useful properties. Some of these are shown in Theorem 8.1.

P R O O F To prove the first property, let Then and

The second and third properties follow directly from the first. Finally, the fourth propertyfollows the definition of the complex conjugate. That is,

Find the product of and its complex conjugate.

S O L U T I O N Because you have

zz � �1 � 2i��1 � 2i� � 12 � 22 � 1 � 4 � 5.

z � 1 � 2i,

z � 1 � 2i

E X A M P L E 2 Finding the Product of Complex Conjugates

� a � bi � a � bi � z.� �a � bi��z�

zz � �a � bi��a � bi� � a2 � abi � abi � b2i2 � a2 � b2.

z � a � biz � a � bi.

Realaxis2 3−2

−2−3−4−5

−3 5 6 7

12345

z = 4 − 5i

z = 4 + 5i

Imaginary axis

Realaxis

z = −2 − 3i

z = −2 + 3i

2

−2

−3

−3−4 −1 1 2

3

Imaginaryaxis

For a complex number the following properties are true.

1.2.3. if and only if 4. �z� � z

z � 0.zz � 0zz � 0zz � a2 � b2

z � a � bi,THEOREM 8.1

Properties of Complex Conjugates

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The Modulus of a Complex Number

Because a complex number can be represented by a vector in the complex plane, it makessense to talk about the length of a complex number. This length is called the modulus ofthe complex number.

R E M A R K : The modulus of a complex number is also called the absolute value of thenumber. In fact, when is a real number, you have

For and determine the value of each modulus.

(a) (b) (c)

S O L U T I O N (a)

(b)

(c) Because you have

Note that in Example 3, In Exercise 40, you are asked to prove that this multiplicative property of the modulus always holds. Theorem 8.2 states that the modulusof a complex number is related to its conjugate.

P R O O F Let then and you have

zz � �a � bi��a � bi� � a2 � b2 � �z�2.

z � a � biz � a � bi,

�zw� � �z� �w�.

�zw� � �152 � 162 � �481.

zw � �2 � 3i��6 � i� � 15 � 16i,�w� � �62 � ��1�2 � �37�z� � �22 � 32 � �13

�zw��w��z�w � 6 � i,z � 2 � 3i

E X A M P L E 3 Finding the Modulus of a Complex Number

�z� � �a2 � 02 � �a�.z � a � 0i

The modulus of the complex number is denoted by and is given by

�z� � �a2 � b2.

�z�z � a � biDefinition of the Modulus of a

Complex Number

For a complex number

�z�2 � zz.

z,THEOREM 8.2

The Modulus of a Complex Number

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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

and and assume that and are not both 0. If the quotient

is to make sense, it has to be true that

But, because you can form the linear system below.

Solving this system of linear equations for and yields

and

Now, because the definition below isobtained.

R E M A R K : If then and In other words, as is the case with real numbers, division of complex numbers by zero is not defined.

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.

�ac � bd

c2 � d2�

bc � ad

c2 � d2i

��ac � bd� � �bc � ad�i

c2 � d 2

a � bi

c � di�

a � bi

c � di �c � di

c � di� ��a � bi��c � di��c � di��c � di�

w � 0.c � d � 0,c2 � d2 � 0,

zw � �a � bi��c � di� � �ac � bd� � �bc � ad�i,

y �bc � ad

ww.x �

ac � bd

ww

yx

dx � cy � b

cx � dy � a

z � a � bi,

z � w�x � yi� � �c � di��x � yi� � �cx � dy� � �dx � cy�i.

zw

� x � yi

dcw � c � diz � a � bi

The quotient of the complex numbers and is defined as

provided c2 � d2 � 0.

z

w�

a � bi

c � di�

ac � bd

c2 � d2�

bc � ad

c2 � d2i �

1

�w�2 �zw �

w � c � diz � a � biDefinition of Division of

Complex Numbers

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(a)

(b)

Now that you can divide complex numbers, you can find the (multiplicative) inverse ofa complex matrix, as demonstrated in Example 5.

Find the inverse of the matrix

and verify your solution by showing that

S O L U T I O N Using the formula for the inverse of a matrix from Section 2.3, you have

Furthermore, because

you can write

To verify your solution, multiply and as follows.

� �10

01 �

110�

100

010

AA�1 � �2 � i3 � i

�5 � 2i�6 � 2i

110

��20�10

17 � i7 � i

A�1A

�110�

�20�10

17 � i7 � i.

�1

3 � i�1

3 � i��6 � 2i��3 � i��3 � i��3 � i�

�5 � 2i��3 � i��2 � i��3 � i�

A�1 �1

3 � i ��6 � 2i�3 � i

5 � 2i2 � i

� 3 � i

� ��12 � 6i � 4i � 2� � ��15 � 6i � 5i � 2� �A� � �2 � i��6 � 2i� � ��5 � 2i��3 � i�

A�1 �1

�A� ��6 � 2i�3 � i

5 � 2i2 � i.

2 � 2

AA�1 � I2.

A � �2 � i

3 � i

�5 � 2i

�6 � 2i,

E X A M P L E 5 Finding the Inverse of a Complex Matrix

2 � i

3 � 4i�

2 � i

3 � 4i�3 � 4i

3 � 4i� �2 � 11i

9 � 16�

2

25�

11

25i

1

1 � i�

1

1 � i �1 � i

1 � i� �1 � i

12 � i2�

1 � i

2�

1

2�

1

2i

E X A M P L E 4 Division of Complex Numbers

332600_08_2.qxp 4/17/08 11:29 AM Page 495

ExercisesSECTION 8.2In Exercises 1–6, find the complex conjugate and graphically represent both and

1. 2.

3. 4.

5. 6.

In Exercises 7–12, find the indicated modulus, whereand

7. 8. 9.

10. 11. 12.

13. Verify that where and

14. Verify that where and

In Exercises 15–20, perform the indicated operations.

15. 16.

17. 18.

19. 20.3 � i

�2 � i��5 � 2i��2 � i��3 � i�

4 � 2i

5 � i

4 � i

3 � �2 i

3 � �2 i

1

6 � 3i

2 � i

i

v � �2 � 3i.z � 1 � 2i�zv2� � �z��v2� � �z��v�2,

w � �1 � 2i.z � 1 � i�wz� � �w��z� � �zw�,

�zv2��v��wz��zw��z2��z�

v � �5i.w � �3 � 2i,z � 2 � i,

z � �3z � 4

z � 2iz � �8i

z � 2 � 5iz � 6 � 3i

z.zz


The last theorem in this section summarizes some useful properties of complex conjugates.

P R O O F To prove the first property, let and Then

The proof of the second property is similar. The proofs of the other two properties areleft to you.

� z � w.

� �a � bi� � �c � di� � �a � c� � �b � d �i

z � w � �a � c� � �b � d�i

w � c � di.z � a � bi

If your computer software program or graphing utility can perform operations with complex matrices, then you can verify the result of Example 5. If you have matrix stored on a graphing utility, evaluate A�1.

ATechnology

Note

For the complex numbers and the following properties are true.

1.2.3.4. zw � zw

zw � z wz � w � z � wz � w � z � w

w,zTHEOREM 8.3

Properties of Complex Conjugates

332600_08_2.qxp 4/17/08 11:29 AM Page 496

In Exercises 21–24, perform the operation and write the result instandard form.

21. 22.

23. 24.

In Exercises 25–28, use the given zero to find all zeros of the polynomial function.

25. Zero:

26. Zero:

27. Zero:

28. Zero:

In Exercises 29 and 30, find each power of the complex number

(a) (b) (c) (d)

29.

30.

In Exercises 31–36, determine whether the complex matrix has aninverse. If is invertible, find its inverse and verify that

31. 32.

33. 34.

35. 36.

In Exercises 37 and 38, determine all values of the complex numberfor which is singular. (Hint: Set and solve for )

37.

38.

39. Prove that if and only if is real.

40. Prove that for any two complex numbers and each of thestatements below is true.

(a)(b) If then

41. Describe the set of points in the complex plane that satisfieseach of the statements below.

(a)(b)(c)(d)

True or False? In Exercises 42 and 43, determine whether eachstatement 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 casesor cite an appropriate statement from the text.

42.

43. There is no complex number that is equal to its complex conjugate.

44. Describe the set of points in the complex plane that satisfieseach of the statements below.

(a)(b)(c)(d)

45. (a) Evaluate for n � 1, 2, 3, 4, and 5.

(b) Calculate and

(c) Find a general formula for for any positive integer

46. (a) Verify that

(b) Find the two square roots of

(c) Find all zeros of the polynomial x 4 � 1.

i.

�1 � i�2 �2

� i.

n.�1i�n

�1i�2010.�1i�2000

�1i�n

�z� > 3�z � 1� � 1�z � i� � 2�z� � 4

i � i 2 � 0

2 � �z� � 5�z � i� � 2�z � 1 � i� � 5�z� � 3

�zw� � �z��w�.w � 0,�zw� � �z��w�

w,z

zz � z

A � �2

1 � i1

2i�1 � i

0

1 � iz0

A � � 53i

z2 � i

z.det�A� � 0Az

A � �i00

0i0

00iA � �

100

01 � i

0

00

1 � i

A � �1 � i0

21 � iA � �1 � i

12

1 � i

A � �2i3

�2 � i3iA � � 6

2 � i3ii

AA�1 � I.AA

z � 1 � i

z � 2 � i

z�2z�1z 3z 2

z.

�1 � 3ip�x� � x3 � 4x2 � 14x � 20

�3 � �2 ip�x� � x4 � 3x3 � 5x2 � 21x � 22

�3 � ip�x� � 4x3 � 23x2 � 34x � 10

1 � �3 ip�x� � 3x3 � 4x2 � 8x � 8

1 � i

i�

3

4 � i

i

3 � i�

2i

3 � i

2i

2 � i�

5

2 � i

2

1 � i�

3

1 � i


332600_08_2.qxp 4/17/08 11:30 AM Page 497


Polar Form and DeMoivre’s Theorem

At this point you can add, subtract, multiply, and divide complex numbers. However, thereis still one basic procedure that is missing from the algebra of complex numbers. To seethis, consider the problem of finding the square root of a complex number such as Whenyou use the four basic operations (addition, subtraction, multiplication, and division), thereseems to be no reason to guess that

That is,

To work effectively with powers and roots of complex numbers, it is helpful to use a polarrepresentation for complex numbers, as shown in Figure 8.6. Specifically, if is anonzero complex number, then let be the angle from the positive -axis to the radial linepassing through the point and let be the modulus of So,

and

and you have from which the polar form of a complexnumber is obtained.

R E M A R K : The polar form of is expressed as where isany angle.

Because there are infinitely many choices for the argument, the polar form of a complexnumber 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( ). Two nonzero complex numbersin polar form are equal if and only if they have the same modulus and the same principalargument.

Find the polar form of each of the complex numbers. (Use the principal argument.)

(a) (b) (c) z � iz � 2 � 3iz � 1 � i

E X A M P L E 1 Finding the Polar Form of a Complex Number

z

�� < � � �

��

�z � 0�cos � � i sin ��,z � 0

a � bi � �r cos �� r sin ��i,

r � �a2 � b2b � r sin �,a � r cos �,

a � bi.r�a, b�x�

a � bi

�1 � i�2 �2

� i.�i �1 � i�2

.

i.

8.3

Imaginaryaxis

Realaxis

br

a 0

θ

θ

Complex Number: a + biRectangular Form: (a, b) Polar Form: (r, )

(a, b)

Figure 8.6

The polar form of the nonzero complex number is given by

where and The number is themodulus of and is the argument of z.�z

rtan � � b�a.a � r cos �, b � r sin �, r � �a2 � b2,

z � r�cos � � i sin ��

z � a � biDefinition of the Polar Form of a

Complex Number

332600_08_3.qxp 4/17/08 11:30 AM Page 498

Sect ion 8 .3 Polar Form and DeMoivre ’s Theorem 499

S O L U T I O N (a) Because and which implies that From and you have

and

So, and

(b) Because and then which implies that So,

and

and it follows that So, the polar form is

(c) Because and it follows that and so

The polar forms derived in parts (a), (b), and (c) are depicted graphically in Figure 8.7.

Figure 8.7

Express the complex number in standard form.

z � 8�cos��

3� � i sin��

3�

E X A M P L E 2 Converting from Polar to Standard Form

Imaginaryaxis

Realaxis

θ

z = i

π2

π2

1

( )(c) z = 1 cos + i sin

Realaxis1 2

4

3

2

1θ

z = 2 + 3i

Imaginaryaxis

(b) z ≈ 13[cos(0.98)+ i sin(0.98)]

Imaginaryaxis

Realaxis

[ ) ]π4

−1

−2

θ

z = 1 − i

(a) z = 2 cos − + i sin − ( )π4(

z � 1�cos �

2� i sin

�

2�.

� � ��2,r � 1b � 1,a � 0

z �13 �cos�0.98� � i sin�0.98�.

� 0.98.

sin � �b

r�

3�13

cos � �a

r�

2�13

r � �13.r2 � 22 � 32 � 13,b � 3,a � 2

z � �2 �cos ��

4� � i sin��

4�.

� � ��4

sin � �b

r� �

1�2

� ��2

2 .cos � �

a

r�

1�2

��2

2

b � r sin �,a � r cos �r � �2.b � �1, then r2 � 12 � ��1�2 � 2,a � 1

332600_08_3.qxp 4/17/08 11:30 AM Page 499


S O L U T I O N Because and you can obtain the standard form

The polar form adapts nicely to multiplication and division of complex numbers.Suppose you have two complex numbers in polar form

and

Then the product of and is expressed as

Using the trigonometric identities

and

you have

This establishes the first part of the next theorem. The proof of the second part is left toyou. (See Exercise 65.)

This theorem says that to multiply two complex numbers in polar form, multiply moduliand add arguments. To divide two complex numbers, divide moduli and subtract arguments.(See Figure 8.8.)

z1z2 � r1r2 �cos��1 � �2� � i sin��1 � �2��.

sin ��1 � �2� � sin �1 cos �2 � cos �1 sin �2

cos��1 � �2� � cos �1 cos �2 � sin �1 sin �2

� r1r2 ��cos �1 cos �2 � sin �1 sin �2� � i �cos �1 sin �2 � sin �1 cos �2��. z1z2 � r1r2�cos �1 � i sin �1��cos �2 � i sin �2�

z2z1

z2 � r2�cos �2 � i sin �2�.z1 � r1�cos �1 � i sin �1�

z � 8�cos��

3� � i sin��

3� � 8�12

� i�32 � 4 � 4�3i.

sin��3� � ��3�2,cos��3� � 1�2

Given two complex numbers in polar form

and

the product and quotient of the numbers are as follows.

Product

Quotientz1z2

�r1r2

�cos��1 � �2� � i sin��1 � �2��, z2 � 0

z1z2 � r1r2 �cos��1 � �2� � i sin��1 � �2��

z2 � r2�cos �2 � i sin �2�z1 � r1�cos �1 � i sin �1�

THEOREM 8.4

Product and Quotient of Two Complex Numbers

332600_08_3.qxp 4/17/08 11:30 AM Page 500


Figure 8.8

Find and for the complex numbers

and

S O L U T I O N Because you have the polar forms of and you can apply Theorem 8.4, as follows.

R E M A R K : Try performing the multiplication and division in Example 3 using the standard forms

and z2 ��36

�16

i.z1 �5�2

2�

5�2

2 i

z1

z2

�5

1�3 �cos��

4�

�

6� � i sin��

4�

�

6� � 15�cos �

12� i sin

�

12�

z1z2 � �5��1

3� �cos��

4�

�

6� � i sin��

4�

�

6� �5

3 �cos

5�

12� i sin

5�

12�

z2,z1

z2 �13�cos

�

6� i sin

�

6�.z1 � 5�cos �

4� i sin

�

4�z1�z2z1z2

E X A M P L E 3 Multiplying and Dividing in Polar Form

Imaginaryaxis

Realaxis

z1z2

r1r2

r2

r1

z1 z2

θ1θ1

θ 2

θ 2−

1 2To divide z and z :Divide moduli and subtract arguments.

Imaginaryaxis

Realaxis

z1z2

r1r2

r2r1

z1

z2

θ1

θ1

θ 2

θ 2+

1 2To multiply z and z :Multiply moduli and add arguments.

multiply

add add

subtract subtract

divide

332600_08_3.qxp 4/17/08 11:30 AM Page 501


DeMoivre’s Theorem

The final topic in this section involves procedures for finding powers and roots of complexnumbers. Repeated use of multiplication in the polar form yields

Similarly,

This pattern leads to the next important theorem, named after the French mathematicianAbraham DeMoivre (1667–1754). You are asked to prove this theorem in Review Exercise 85.

Find and write the result in standard form.

S O L U T I O N First convert to polar form. For

and

which implies that So,

By DeMoivre’s Theorem,

� 4096�1 � i�0�� 4096.

� 4096�cos 8� � i sin 8��

� 212�cos 12�2��

3� i sin

12�2��3

��1 � �3 i�12� �2�cos

2�

3� i sin

2�

3 �12

�1 � �3 i � 2�cos 2�

3� i sin

2�

3 �.

� � 2��3.

tan � ��3

�1� ��3r � ��1�2 � ��3�2 � 2

�1 � �3 i,

��1 � �3 i�12

E X A M P L E 4 Raising a Complex Number to an Integer Power

z5 � r 5�cos 5� � i sin 5��. z4 � r4�cos 4� � i sin 4��

z3 � r �cos � � i sin ��r2 �cos 2� � i sin 2�� r3�cos 3� � i sin 3��. z2 � r �cos � � i sin ��r �cos � � i sin �� r2�cos 2� � i sin 2�� z � r �cos � � i sin ��

If and is any positive integer, then

zn � rn�cos n� � i sin n��.

nz � r�cos � � i sin ��THEOREM 8.5

DeMoivre’s Theorem

332600_08_3.qxp 4/17/08 11:30 AM Page 502


Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomialof degree has zeros in the complex number system. So, a polynomial such as

has six zeros, and in this case you can find the six zeros by factoring andusing the Quadratic Formula.

Consequently, the zeros are

and

Each of these numbers is called a sixth root of 1. In general, the th root of a complexnumber is defined as follows.

DeMoivre’s Theorem is useful in determining roots of complex numbers. To see howthis is done, let be an th root of where

and

Then, by DeMoivre’s Theorem you have and because it follows that

Now, because the right and left sides of this equation represent equal complex numbers, youcan equate moduli to obtain which implies that and equate principal arguments to conclude that and must differ by a multiple of Note that is a positive real number and so is also a positive real number. Consequently, for someinteger which implies that

Finally, substituting this value of into the polar form of produces the result stated in thenext theorem.

w

�� 2�k

n .

k, n � � � 2�k,s � n�r

r2�.n�s � n�r ,sn � r,

sn �cos n � i sin n� � r�cos � � i sin ��.

wn � z,wn � sn�cos n � i sin n�,

z � r �cos � � i cos ��.w � s�cos � i sin �

z,nw

n

x �1 ± �3 i

2.x �

�1 ± �3i2

,x � ±1,

x6 � 1 � �x3 � 1��x3 � 1� � �x � 1��x2 � x � 1��x � 1��x2 � x � 1�

p�x� � x6 � 1nn

The complex number is an th root of the complex number if

z � wn � �a � bi�n .

znw � a � biDefinition of the nth Root of a Complex Number

For any positive integer the complex number has exactly distinct roots. These roots are given by

where k � 0, 1, 2, . . . , n � 1.

n�r �cos�� 2�kn � � i sin�� 2�k

n �nn

z � r �cos � � i sin ��n,THEOREM 8.6

The nth Roots of a Complex Number

332600_08_3.qxp 4/17/08 11:30 AM Page 503


R E M A R K : Note that when exceeds the roots begin to repeat. For instance, ifthe angle is

which yields the same values for the sine and cosine as

The formula for the th roots of a complex number has a nice geometric interpretation,as shown in Figure 8.9. Note that because the th roots all have the same modulus (length)

they will lie on a circle of radius with center at the origin. Furthermore, the rootsare equally spaced around the circle, because successive th roots have arguments thatdiffer by

You have already found the sixth roots of 1 by factoring and the Quadratic Formula. Trysolving the same problem using Theorem 8.6 to see if you get the roots shown in Figure8.10. When Theorem 8.6 is applied to the real number 1, the th roots have a specialname—the th roots of unity.

Figure 8.10

Determine the fourth roots of

S O L U T I O N In polar form, you can write as

so that Then, by applying Theorem 8.6, you have

� cos��

8�

k�

2 � � i sin��

8�

k�

2 �.

i1�4 � 4�1 �cos��24

�2k�

4 � � i sin��24

�2k�

4 �r � 1 and � � ��2.

i � 1�cos �

2� i sin

�

2�i

i.

E X A M P L E 5 Finding the nth Roots of a Complex Number

Imaginaryaxis

Realaxis

The Sixth Roots of Unity

1−1

1

1

1

1

2

2

2

2

2

2

2

2

3

3

3

3

+

−

+

−

i

i

i

i

−

−

nn

2��n.n

nn�rn�r,n

n

k � 0.

� � 2�n

n�

�

n� 2�

k � n,n � 1,k

rn

2π

2π

n

n

The nth Roots of aComplex Number

Realaxis

Imaginaryaxis

Figure 8.9

332600_08_3.qxp 4/17/08 11:30 AM Page 504

In Exercises 1–4, express the complex number in polar form.

1. 2.

3. 4. Imaginaryaxis

Realaxis

3i

1

1−1

2

3

Imaginaryaxis

Realaxis

−6 1

2

−2

−2−3−4−5−6

−3

3Imaginary

axis

Realaxis

1 + 3i

1 2

1

2

3

−1

Imaginaryaxis

Realaxis

1

−1

−2

2

2 − 2i

ExercisesSECTION 8.3


Setting and 3, you obtain the four roots

as shown in Figure 8.11.

R E M A R K : In Figure 8.11, note that when each of the four angles andis multiplied by 4, the result is of the form

Figure 8.11

Imaginaryaxis

Realaxis

13π

π8

π8

9π8

813π

8

9π8cos + i sin

5π8

5π8cos + i sin

cos + i sin

cos + i sin

��2� � 2k�.13��8��8, 5��8, 9��8,

z4 � cos 13�

8� i sin

13�

8

z3 � cos 9�

8� i sin

9�

8

z2 � cos 5�

8� i sin

5�

8

z1 � cos �

8� i sin

�

8

k � 0, 1, 2,

332600_08_3.qxp 4/17/08 11:30 AM Page 505


In Exercises 5–16, represent the complex number graphically, andgive the polar form of the number. (Use the principal argument.)

5. 6.

7. 8.

9. 10.

11. 7 12. 4

13. 14.

15. 16.

In Exercises 17–26, represent the complex number graphically, andgive the standard form of the number.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

In Exercises 27–34, perform the indicated operation and leave theresult in polar form.

27.

28.

29.

30.

31.

32.

33.

34.

In Exercises 35–44, use DeMoivre’s Theorem to find the indicatedpowers of the complex number. Express the result in standard form.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

In Exercises 45–56, (a) use DeMoivre’s Theorem to find the indicated roots, (b) represent each of the roots graphically, and (c) express each of the roots in standard form.

45. Square roots:

46. Square roots:

47. Fourth roots:

48. Fifth roots:

49. Square roots:

50. Fourth roots:

51. Cube roots:

52. Cube roots:

53. Cube roots: 8 54. Fourth roots: i

55. Fourth roots: 1 56. Cube roots: 1000

In Exercises 57–64, find all the solutions to the equation and represent your solutions graphically.

57. 58.

59. 60.

61. 62.

63. 64.

65. When provided with two complex numbersand

with prove that

z 1z 2

�r1r2

�cos��1 � �2� � i sin��1 � �2��.

z 2 � 0,z 2 � r2�cos �2 � i sin �2�,z 1 � r1�cos �1 � i sin �1�

x 4 � i � 0x 3 � 64i � 0

x 4 � 81 � 0x 5 � 243 � 0

x3 � 27 � 0x 3 � 1 � 0

x4 � 16i � 0x 4 � i � 0

�4�2 �1 � i��

1252 �1 � �3i�

625i

�25i

32�cos 5�

6� i sin

5�

6 �

16�cos 4�

3� i sin

4�

3 �

9�cos 2�

3� i sin

2�

3 �

16�cos �

3� i sin

�

3�

�5�cos 3�

2� i sin

3�

2 �4�2�cos �

2� i sin

�

2�8

�cos 5�

4� i sin

5�

4 �10�3�cos 5�

6� i sin

5�

6 �4

�5�cos �

9� i sin

�

9�3

�1 � �3i�3

��3 � i�7��1 � i�10

�2 � 2i�6�1 � i�4

9�cos�3��4� � i sin�3��4��5�cos��4� � i sin��4��

12�cos��3� � i sin��3��3�cos��6� � i sin��6��

cos�5��3� � i sin�5��3�cos� � i sin�

2�cos�2��3� � i sin�2��3��4[cos�5��6� � i sin�5��6�]

�3�cos �

3� i sin

�

3��1

3 �cos

2�

3� i sin

2�

3 ��0.5�cos� � i sin�� 0.5�cos�� i sin��

�3

4 �cos

�

2� i sin

�

2��6�cos �

4� i sin

�

4�

�3�cos �

3� i sin

�

3��4�cos �

6� i sin

�

6�

6�cos � � i sin ��7�cos 0 � i sin 0�

6�cos 5�

6� i sin

5�

6 �4�cos 3�

2� i sin

3�

2 �

8�cos �

6� i sin

�

6�3.75�cos �

4� i sin

�

4�

3

4 �cos

7�

4� i sin

7�

4 �3

2 �cos 5�

3� i sin

5�

3 �

5�cos 3�

4� i sin

3�

4 �2�cos �

2� i sin

�

2�

5 � 2i�1 � 2i

2�2 � i3 � �3 i

�2i6i

52��3 � i ��2�1 � �3i �2 � 2i�2 � 2i

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66. Show that the complex conjugate of is

67. Use the polar forms of and in Exercise 66 to find each of thefollowing.

(a)

(b)

68. Show that the negative of is

69. Writing

(a) Let

Sketch and in the complex plane.

(b) What is the geometric effect of multiplying a complex number by What is the geometric effect of dividing

by

70. Calculus Recall that the Maclaurin series for sin and cos are

(a) Substitute in the series for and show that

(b) Show that any complex number can be expressed in polar form as

(c) Prove that if then

(d) Prove the amazing formula


71. Although the square of the complex number is given bythe absolute value of the complex number

is defined as

72. Geometrically, the th roots of any complex number are allequally spaced around the unit circle centered at the origin.

zn a � bi � �a2 � b2.z � a � bi

�bi�2 � �b2,bi

ei� � �1.

z � re�i�.z � rei�,

z � rei�.z � a � bi

ei� � cos � � i sin �.exx � i�

cos x � 1 �x 2

2!�

x 4

4!�

x6

6!� . . . .

sin x � x �x 3

3!�

x5

5!�

x 7

7!� . . .

ex � 1 � x �x 2

2!�

x3

3!�

x 4

4!� . . .

xx,e x,

i?zi?z

z�iiz,z,

z � r�cos � � i sin �� 2�cos �

6� i sin

�

6�.

�z � r�cos�� i sin�� .z � r�cos � � i sin ��

z�z, z � 0

zz

zz

z � r �cos�� i sin��.z � r�cos � � i sin ��

332600_08_3.qxp 4/17/08 11:30 AM Page 507

Sect ion 8 .4 Complex Vector Spaces and Inner Products 509

Complex Vector Spaces and Inner Products

All the vector spaces you have studied so far in the text have been real vector spacesbecause the scalars have been real numbers. A complex vector space is one in which thescalars are complex numbers. So, if are vectors in a complex vector space,then a linear combination is of the form

where the scalars are complex numbers. The complex version of is thecomplex vector space consisting of ordered -tuples of complex numbers. So, a vectorin has the form

It is also convenient to represent vectors in by column matrices of the form

As with the operations of addition and scalar multiplication in are performed component by component.

Let

and

be vectors in the complex vector space Determine each vector.

(a) (b) (c)

S O L U T I O N (a) In column matrix form, the sum is

(b) Because and you have

(c)

� �12 � i, �11 � i� � �3 � 6i, 9 � 3i� � ��9 � 7i, 20 � 4i�

3v � �5 � i�u � 3�1 � 2i, 3 � i� � �5 � i��2 � i, 4�

�2 � i�v � �2 � i��1 � 2i, 3 � i � � �5i, 7 � i�.

�2 � i��3 � i� � 7 � i,�2 � i��1 � 2i� � 5i

v � u � �1 � 2i3 � i� � ��2 � i

4� � ��1 � 3i7 � i�.

v � u

3v � �5 � i�u�2 � i�vv � u

C2.

u � ��2 � i, 4�v � �1 � 2i, 3 � i�

E X A M P L E 1 Vector Operations in C n

CnRn,

v � �a1 � b1ia2 � b2i...an � bni

�.

Cn

v � �a1 � b1i, a2 � b2i, . . . , an � bni�.

CnnCn

Rnc1, c2, . . . , cm

c1v1 � c2v2 � � � � � cmvm

v1, v2, . . . , vm

8.4

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Many of the properties of are shared by For instance, the scalar multiplicativeidentity is the scalar 1 and the additive identity in is The standardbasis for is simply

which is the standard basis for Because this basis contains vectors, it follows that the dimension of is Other bases exist; in fact, any linearly independent set of vectors in

can be used, as demonstrated in Example 2.

Show that

is a basis for

S O L U T I O N Because has a dimension of 3, the set will be a basis if it is linearly indepen-dent. To check for linear independence, set a linear combination of the vectors in equal to

as follows.

This implies that

So, and you can conclude that is linearly independent.

Use the basis in Example 2 to represent the vector

v � �2, i, 2 � i�.

S

E X A M P L E 3 Representing a Vector in C n by a Basis

�v1, v2, v3�c1 � c2 � c3 � 0,

c3i � 0.

c2i � 0

�c1 � c2�i � 0

��c1 � c2�i, c2i, c3i� � �0, 0, 0� �c1i , 0, 0� � �c2i, c2i, 0� � �0, 0, c3i� � �0, 0, 0�

c1v1 � c2v2 � c3v3 � �0, 0, 0�

0,S

�v1, v2, v3�C 3

C 3.

S � ��i, 0, 0�, �i, i, 0�, �0, 0, i��

E X A M P L E 2 Verifying a Basis

Cnnn.Cn

nRn.

en � �0, 0, 0, . . . , 1�

.

.

.e2 � �0, 1, 0, . . . , 0�e1 � �1, 0, 0, . . . , 0�

Cn0 � �0, 0, 0, . . . , 0�.Cn

Cn.Rn

v1 v2 v3

332600_08_4.qxp 4/17/08 11:31 AM Page 510


S O L U T I O N By writing

you can obtain

which implies that

and

So,

Try verifying that this linear combination yields

Other than there are several additional examples of complex vector spaces. For instance, the set of complex matrices with matrix addition and scalar multiplicationforms a complex vector space. Example 4 describes a complex vector space in which thevectors are functions.

Consider the set of complex-valued functions of the form

where and are real-valued functions of a real variable. The set of complex numbersform the scalars for and vector addition is defined by

It can be shown that scalar multiplication, and vector addition form a complex vectorspace. For instance, to show that is closed under scalar multiplication, let bea complex number. Then

is in S.

� �af1�x� � bf2�x� � i�bf1�x� � af2�x� cf�x� � �a � bi��f1�x� � if2�x�

c � a � biSS,

� �f1�x� � g1�x� � i�f2�x� � g2�x�. f�x� � g�x� � �f1�x� � if2�x� � �g1(x� � i g2�x�

S,f2f1

f�x� � f1�x� � if2�x�

S

E X A M P L E 4 The Space of Complex-Valued Functions

m � nCn,

�2, i, 2 � i�.

v � ��1 � 2i�v1 � v2 � ��1 � 2i�v3.

c3 �2 � i

i� �1 � 2i .c1 �

2 � i

i� �1 � 2i,

c2 � 1,

c3i � 2 � i

c2i � i

�c1 � c2�i � 2

� �2, i, 2 � i�, � ��c1 � c2�i, c2i, c3i�

v � c1v1 � c2v2 � c3v3

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The definition of the Euclidean inner product in is similar to the standard dot product in except that here the second factor in each term is a complex conjugate.

R E M A R K : Note that if and happen to be “real,” then this definition agrees with thestandard inner (or dot) product in .

Determine the Euclidean inner product of the vectors

and

S O L U T I O N

Several properties of the Euclidean inner product are stated in the following theorem.

P R O O F The proof of the first property is shown below, and the proofs of the remaining propertieshave been left to you. Let

and v � �v1, v2, . . . , vn �.u � �u1, u2, . . . , un �

Cn

� 3 � i

� �2 � i��1 � i� � 0�2 � i� � �4 � 5i��0� u � v � u1v1 � u2v2 � u3v3

v � �1 � i, 2 � i, 0�.u � �2 � i, 0, 4 � 5i�

E X A M P L E 5 Finding the Euclidean Inner Product in C 3

Rnvu

Rn,Cn

Let and be vectors in The Euclidean inner product of and is given by

u � v � u1v1 � u2v2 � � � � � unvn.

vuCn.vuDefinition of the EuclideanInner Product in Cn

Let and be vectors in and let be a complex number. Then the following properties are true.

1.2.3.4.5.6. if and only if u � 0.u � u � 0

u � u � 0u � �kv� � k�u � v��ku� � v � k�u � v��u � v� � w � u � w � v � wu � v � v � u

kCnwu, v,THEOREM 8.7

Properties of the Euclidean Inner Product

332600_08_4.qxp 4/17/08 11:31 AM Page 512


Then

You will now use the Euclidean inner product in to define the Euclidean norm (orlength) of a vector in and the Euclidean distance between two vectors in .

The Euclidean norm and distance may be expressed in terms of components as

Determine the norms of the vectors

and

and find the distance between and

S O L U T I O N The norms of and are expressed as follows.

� �2 � 5 � 0�12 � �7

� ��12 � 12� � �22 � 12� � �02 � 02�12

�v� � � v1 2 � v2 2 � v3 2�12

� �5 � 0 � 41�12 � �46

� ��22 � 12� � �02 � 02� � �42 � 52�12

�u� � � u1 2 � u2 2 � u3 2�12

vu

v.u

v � �1 � i, 2 � i, 0�u � �2 � i, 0, 4 � 5i�

E X A M P L E 6 Finding the Euclidean Norm and Distance in Cn

d�u, v� � � u1 � v1 2 � u2 � v2 2 � . . . � un � vn 2�12.

�u� � � u1 2 � u2 2 � . . . � un 2�12

CnCnCn

� u � v. � u1v1 � u2v2 � � � � � unvn

� v1u1 � v2u2 � . . . � vnun

� v1u1 � v2u2 � . . . � vnun

v � u � v1u1 � v2u2 � . . . � vnun

The Euclidean norm (or length) of in is denoted by and is

The Euclidean distance between and is

d�u, v� � �u � v� .

vu

�u� � �u � u�12.

�u�CnuDefinitions of the Euclidean Norm

and Distance in Cn

332600_08_4.qxp 4/17/08 11:31 AM Page 513


The distance between and is expressed as

Complex Inner Product Spaces

The Euclidean inner product is the most commonly used inner product in . On occasion,however, it is useful to consider other inner products. To generalize the notion of an innerproduct, use the properties listed in Theorem 8.7.

A complex vector space with a complex inner product is called a complex inner productspace or unitary space.

Let and be vectors in the complex space Show that the function defined by

is a complex inner product.

S O L U T I O N Verify the four properties of a complex inner product as follows.

1.2.

3.4.

Moreover, if and only if

Because all properties hold, is a complex inner product.�u, v�u1 � u2 � 0.�u, u� � 0

�u, u� � u1u1 � 2u2u2 � u1 2 � 2 u2 2 � 0�ku, v� � �ku1�v1 � 2�ku2�v2 � k�u1v1 � 2u2v2 � � k �u, v�

� �u, w� � �v, w� � �u1w1 � 2u2w2 � � �v1w1 � 2v2w2 �

�u � v, w� � �u1 � v1�w1 � 2�u2 � v2�w2

� u1v1 � 2u2v2 � �u, v�v1u1 � 2v2u2�v, u� �

�u, v� � u1v1 � 2u2v2

C2.v � �v1, v2�u � �u1, u2�

E X A M P L E 7 A Complex Inner Product Space

Cn

� �1 � 5 � 41�12 � �47.

� ��12 � 02� � ��2�2 � ��1�2� � �42 � ��5�2�12

� ��1, �2 � i, 4 � 5i�� d�u, v� � �u � v�

vu

Let and be vectors in a complex vector space. A function that associates and withthe complex number is called a complex inner product if it satisfies the followingproperties.

1.2.3.4. and if and only if u � 0.�u, u� � 0�u, u� � 0

�ku, v� � k�u, v��u � v, w� � �u, w� � �v, w��u, v� � �v, u�

�u, v�vuvuDefinition of a

Complex Inner Product

332600_08_4.qxp 4/17/08 11:31 AM Page 514


ExercisesSECTION 8.4In Exercises 1–8, perform the indicated operation using

and

1. 2.

3. 4.

5. 6.

7. 8.

In Exercises 9–12, determine whether is a basis for

9.

10.

11.

12.

In Exercises 13–16, express as a linear combination of each of the following basis vectors.

(a)

(b)

13. 14.

15. 16.

In Exercises 17–24, determine the Euclidean norm of

17. 18.

19. 20.

21. 22.

23.

24.

In Exercises 25–30, determine the Euclidean distance between and

25.

26.

27.

28.

29.

30.

In Exercises 31–34, determine whether the set of vectors is linearlyindependent or linearly dependent.

31.

32.

33.

34.

In Exercises 35–38, determine whether the function is a complexinner product, where and

35.

36.

37.

38.

In Exercises 39–42, use the inner product tofind

39. and

40. and

41. and

42. and

43. Let and If and theset is not a basis for what does this imply about

and

44. Let and Determine a vector suchthat is a basis for

In Exercises 45–49, prove the property, where and are vectors in and is a complex number.

45. 46.

47. 48.

49. if and only if

50. Writing Let be a complex inner product and let be acomplex number. How are and related?

In Exercises 51 and 52, use the inner productwhere

and

to find

51.

52. v � � i3i

�2i�1�u � � 1

1 � i2i0�v � �1

01 � 2i

i�u � �01

i�2i�

�u, v�.

v � �v11

v21

v12

v22�u � �u11

u21

u12

u22�

�u, v� � u11v11 � u12v12 � u21v21 � u22v22

�u, kv��u, v�k�u, v�

u � 0.u � u � 0

u � u � 0u � �kv� � k�u � v��ku� � v � k�u � v��u � v� � w � u � w � v � w

kC n

wu, v,

C 3.�v1, v2, v3�v3v2 � �1, 0, 1�.v1 � �i, i, i�

z3?z1, z2,C3,�v1, v2, v3�

v3 � �z1, z2, z3�v2 � �i, i, 0�.v1 � �i, 0, 0�v � �2 � 3i, �2�u � �4 � 2i, 3�

v � �3 � i, 3 � 2i�u � �2 � i, 2 � i�v � �2 � i, 2i�u � �3 � i, i�

v � �i, 4i�u � �2i, �i�

�u, v�.�u, v� � u1v1 � 2u2v2

�u, v� � u1v1 � u2v2

�u, v� � 4u1v1 � 6u2v2

�u, v� � �u1 � v1� � 2�u2 � v2��u, v� � u1 � u2v2

v � �v1, v2�.u � �u1, u2�

��1 � i, 1 � i, 0�, �1 � i, 0, 0�, �0, 1, 1��1, i, 1 � i�, �0, i, �i�, �0, 0, 1��1 � i, 1 � i, 1�, �i, 0, 1�, ��2, �1 � i, 0��

��1, i�, �i, �1��

u � �1, 2, 1, �2i�, v � �i, 2i, i, 2�u � �1, 0�, v � �0, 1�u � ��2, 2i, �i�, v � �i, i, i�u � �i, 2i, 3i�, v � �0, 1, 0�u � �2 � i, 4, �i�, v � �2 � i, 4, �i�u � �1, 0�, v � �i, i�

v.u

v � �2, �1 � i, 2 � i, 4i�v � �1 � 2i, i, 3i, 1 � i�

v � �0, 0, 0�v � �1, 2 � i, �i�v � �2 � 3i, 2 � 3i�v � 3�6 � i, 2 � i�v � �1, 0�v � �i, �i�

v.

v � �i, i, i�v � ��i, 2 � i, �1�v � �1 � i, 1 � i, �3�v � �1, 2, 0�

��1, 0, 0�, �1, 1, 0�, �0, 0, 1 � i��i, 0, 0�, �i, i, 0�, �i, i, i��

v

S � ��1 � i, 0, 1�, �2, i, 1 � i�, �1 � i, 1, 1��S � ��i, 0, 0�, �0, i, i�, �0, 0, 1��S � ��1, i�, �i, 1��S � ��1, i�, �i, �1��

Cn.S

2iv � �3 � i�w � uu � iv � 2iw

�6 � 3i�v � �2 � 2i�wu � �2 � i�viv � 3w�1 � 2i�w4iw3u

w � �4i, 6�.u � �i, 3 � i�, v � �2 � i, 3 � i�,

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In Exercises 53 and 54, determine the linear transformation that has the given characteristics.

53.

54.

In Exercises 55–58, the linear transformation is shownby Find the image of and the preimage of

55.

56.

57.

58.

59. Find the kernel of the linear transformation from Exercise 55.

60. Find the kernel of the linear transformation from Exercise 56.

In Exercises 61 and 62, find the image of for the indicatedcomposition, where and are the matrices below.

and

61.

62.

63. Determine which of the sets below are subspaces of the vectorspace of complex matrices.

(a) The set of symmetric matrices.(b) The set of matrices satisfying (c) The set of matrices in which all entries are real.(d) The set of diagonal matrices.

64. Determine which of the sets below are subspaces of the vectorspace of complex-valued functions (see Example 4).

(a) The set of all functions satisfying (b) The set of all functions satisfying (c) The set of all functions satisfying


65. Using the Euclidean inner product of and in

66. The Euclidean form of in denoted by is �u � u�2.�u�Cnu

u � v � u1v1 � u2v2 � . . . � unvn.C n,vu

f �i� � f ��i�.f f �0� � 1.f f �i� � 0.f

2 � 22 � 2

�A �T � A.A2 � 22 � 2

2 � 2

T1 � T2

T2 � T1

T2 � ��ii

i�i�T1 � �0

ii0�

T2T1

v � �i, i�

w � �1 � i1 � i

i�v � �

250�,A � �

0i0

1 i i

1�1 0�,

w � �2

2i3i�v � � 2 � i

3 � 2i�,A � �1ii

00i�,

w � �11�v � �

i0

1 � i�,A � �0

ii0

10�,

w � �00�v � �1 � i

1 � i�,A � �1i

0i�,

w.vT�v� � Av.T : Cm → Cn

T�i, 0� � �2 � i, 1�, T�0, i� � �0, �i�T�1, 0� � �2 � i, 1�, T�0, 1� � �0, �i�

T : Cm → Cn

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Sect ion 8 .5 Uni tary and Hermit ian Matr ices 517

Unitary and Hermitian Matrices

Problems involving diagonalization of complex matrices and the associated eigenvalueproblems require the concepts of unitary and Hermitian matrices. These matrices roughlycorrespond to orthogonal and symmetric real matrices. In order to define unitary andHermitian matrices, the concept of the conjugate transpose of a complex matrix must firstbe introduced.

Note that if is a matrix with real entries, then * . To find the conjugate transpose of a matrix, first calculate the complex conjugate of each entry and then take thetranspose of the matrix, as shown in the following example.

Determine * for the matrix

S O L U T I O N

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 inExercises 55–58.

A* � AT � �3 � 7i0

�2i4 � i�

A � �3 � 7i2i

04 � i� � �3 � 7i

�2i0

4 � i�

A � �3 � 7i2i

04 � i�.

A

E X A M P L E 1 Finding the Conjugate Transpose of a Complex Matrix

� ATAA

8.5

The conjugate transpose of a complex matrix denoted by *, is given by

*

where the entries of are the complex conjugates of the corresponding entries of A.A

� A TA

AA,Definition of the Conjugate Transpose of a Complex Matrix

If and are complex matrices and is a complex number, then the following properties are true.

1.2.3.4. �AB�* � B*A*

�kA�* � kA*��A � B�* � A* � B*�A*�* � A

kBATHEOREM 8.8

Properties of the Conjugate Transpose

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Unitary Matrices

Recall that a real matrix is orthogonal if and only if In the complex system,matrices having the property that * are more useful, and such matrices are called unitary.

Show that the matrix is unitary.

S O L U T I O N Because

you can conclude that So, is a unitary matrix.

In Section 7.3, you saw 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

in (a complex Euclidean space) is called orthonormal if the statements below are true.

1.2.

The proof of the next theorem is similar to the proof of Theorem 7.8 presented in Section 7.3.

i � jvi � vj � 0,�vi� � 1, i � 1, 2, . . . , m

Cn

�v1, v2, . . . , vm�

AA* � A�1.

AA* �12�

1 � i1 � i

1 � i1 � i�

12 �

1 � i1 � i

1 � i1 � i� �

14�

40

04� � �1

001� � I2,

A �12 �

1 � i1 � i

1 � i1 � 1�

A

E X A M P L E 2 A Unitary Matrix

A�1 � AA�1 � AT.A

A complex matrix is unitary if

A�1 � A*.

ADefinition of Unitary Matrix

An complex matrix is unitary if and only if its row (or column) vectors form anorthonormal set in Cn.

An � nTHEOREM 8.9

Unitary Matrices

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Show that the complex matrix is unitary by showing that its set of row vectors forms an orthonormal set in

S O L U T I O N Let and be defined as follows.

The length of is

The vectors and can also be shown to be unit vectors. The inner product of and is

Similarly, and So, you can conclude that is an ortho-normal set. Try showing that the column vectors of also form an orthonormal set in C3.�A�

�r1, r2, r3�r2 � r3 � 0.r1 � r3 � 0

� 0. �i

23�

i

23�

1

23�

1

23

� 1

2�i

3� � 1 � i

2 ��i

3� � �1

2�13�

r1 � r2 � 12��

i3� � 1 � i

2 � i3� � �

12�

13�

r2

r1r3r2

� �14

�24

�14�

1�2

� 1.

� �12�

12� � 1 � i

2 �1 � i2 � � �

12��

12��

1�2

� �12�

12� � 1 � i

2 �1 � i2 � � �

12��

12��

1�2

�r1� � �r1 � r1�1�2

r1

r3 � 5i

215,

3 � i

215,

4 � 3i

215 �

r2 � � i3

, i

3,

13�

r1 � 1

2,

1 � i

2, �

1

2�r3r1, r2,

A � �12

�i

35i

215

1 � i2i

33 � i

215

�12

13

4 � 3i

215

�C3.

A

E X A M P L E 3 The Row Vectors of a Unitary Matrix

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Hermitian Matrices

A real matrix is called symmetric if 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 amatrix is called Hermitian after the French mathematician Charles Hermite (1822–1901).

As with symmetric matrices, you can easily recognize Hermitian matrices by inspection.To see this, consider the matrix

The conjugate transpose of has the form

If is Hermitian, then So, you can conclude that must be of the form

Similar results can be obtained for Hermitian matrices of order In other words, asquare matrix is Hermitian if and only if the following two conditions are met.

1. The entries on the main diagonal of are real.2. The entry in the th row and the th column is the complex conjugate of the entry

in the th row and the th column.

Which matrices are Hermitian?

(a) (b)

(c) (d) ��1

23

20

�1

3�1

4��3

2 � i3i

2 � i0

1 � i

�3i1 � i

0�� 0

3 � 2i3 � 2i

4�� 1

3 � i

3 � i

i �

E X A M P L E 4 Hermitian Matrices

ijajijiaij

A

An � n.

A � � a1

b1 � b2ib1 � b2i

d1�.

AA � A*.A

� �a1 � a2ib1 � b2i

c1 � c2id1 � d2i

�.

� �a1 � a2i

b1 � b2i

c1 � c2i

d1 � d2i� A* � AT

A

A � �a1 � a2ic1 � c2i

b1 � b2id1 � d2i

�A.2 � 2

A square matrix is Hermitian if

A � A*.

ADefinition of a Hermitian Matrix

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S O L U T I O N (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 firstcolumn.

(c) This matrix is Hermitian.(d) This matrix is Hermitian because all real symmetric matrices are Hermitian.

One of the most important characteristics of Hermitian matrices is that their eigenvaluesare real. This is formally stated in the next theorem.

P R O O F Let be an eigenvalue of and let

be its corresponding eigenvector. If both sides of the equation are multiplied bythe row vector then

Furthermore, because

it follows that * is a Hermitian matrix. This implies that * is a real number,so is real.

R E M A R K : Note that this theorem implies that the eigenvalues of a real symmetric matrixare real, as stated in Theorem 7.7.

To find the eigenvalues of complex matrices, follow the same procedure as for real matrices.

Avv1 � 1Avv

�v*Av�* � v*A*�v*�* � v*Av,

v*Av � v*�v� � �v*v� � �a12 � b1

2 � a22 � b2

2 � . . . � an2 � bn

2�.

v*,Av � v

v � �a1 � b1i

a2 � b2i

..

.

an � bni�

A

If is a Hermitian matrix, then its eigenvalues are real numbers.ATHEOREM 8.10

The Eigenvalues of aHermitian Matrix

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Find the eigenvalues of the matrix

S O L U T I O N The characteristic polynomial of is

This implies that the eigenvalues of are and

To find the eigenvectors of a complex matrix, use a procedure similar to that used for areal matrix. For instance, in Example 5, the eigenvector corresponding to the eigenvalue

is obtained by solving the following equation.

Using Gauss-Jordan elimination, or a computer software program or graphing utility,obtain the eigenvector corresponding to which is shown below.

Eigenvectors for and can be found in a similar manner. They are

and respectively.�1 � 3i

�2 � i

5�,�

1 � 21i

6 � 9i

13�

3 � �22 � 6

v1 � ��1

1 � 2i

1�

1 � �1,

��4

�2 � i�3i

�2 � i�1

�1 � i

3i�1 � i

�1��v1

v2

v3� � �

000�

� � 3

�2 � i�3i

�2 � i

�1 � i

3i�1 � i

��

v1

v2

v3� � �

000�

� �1

�2.�1, 6,A

� � � 1�� 6�� 2�. � 3 � 32 � 16 � 12

� �3 � 32 � 2 � 6� � �5 � 9 � 3i� � �3i � 9 � 9�� 3i �1 � 3i� � 3i�

� � � 3��2 � 2� � ��2 � i� ��2 � i� � �3i � 3��

�I � A� � � � 3

�2 � i�3i

�2 � i

�1 � i

3i�1 � i

�

A

A � �3

2 � i3i

2 � i0

1 � i

�3i1 � i

0�A.

E X A M P L E 5 Finding the Eigenvalues of a Hermitian Matrix

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Just as you saw in Section 7.3 that real symmetric matrices are orthogonally diagonal-izable, you will now see that Hermitian matrices are unitarily diagonalizable. A squarematrix is unitarily diagonalizable if there exists a unitary matrix such that

is a diagonal matrix. Because is unitary, *, so an equivalent statement is that is unitarily diagonalizable if there exists a unitary matrix such that * is a diagonalmatrix. The next theorem states that Hermitian matrices are unitarily diagonalizable.

P R O O F To prove part 1, let and be two eigenvectors corresponding to the distinct (and real)eigenvalues and Because and you have the equations shownbelow for the matrix product *

* *A* * * *

* * * *

So,

because

and this shows that and are orthogonal. Part 2 of Theorem 8.11 is often called theSpectral Theorem, and its proof is left to you.

The eigenvectors of the Hermitian matrix shown in Example 5 are mutually orthogonal because the eigenvalues are distinct. You can verify this by calculating the Euclidean innerproducts and For example,v2 � v3.v1 � v3,v1 � v2,

E X A M P L E 6 The Eigenvectors of a Hermitian Matrix

v2v1

1� 2, v1*v2 � 0

�2 � 1�v1*v2 � 0

2v1*v2 � 1v1*v2 � 0

v21v2 � 1v1v2 � v1v2 � �1v1��Av1�v22v2 � 2v1Av2 � v1v2 � v1v2 � v1�Av1�

v2.�Av1�Av2 � 2v2,Av1 � 1v12.1

v2v1

APPPAP�1 � PP

P�1AP

PA

Some computer software programs and graphing utilities have built-in programs for finding theeigenvalues and corresponding eigenvectors of complex matrices. For example, on the TI-86, theeigVl key on the matrix math menu calculates the eigenvalues of the matrix and the eigVc keygives the corresponding eigenvectors.

A,

TechnologyNote

If is an Hermitian matrix, then

1. eigenvectors corresponding to distinct eigenvalues are orthogonal.2. is unitarily diagonalizable.A

n � nATHEOREM 8.11

Hermitian Matrices and Diagonalization

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The other two inner products and can be shown to equal zero in a similarmanner.

The three eigenvectors in Example 6 are mutually orthogonal because they correspondto distinct eigenvalues of the Hermitian matrix . Two or more eigenvectors correspondingto the same eigenvalue may not be orthogonal. Once any set of linearly independent eigenvectors is obtained for an eigenvalue, however, the Gram-Schmidt orthonormalizationprocess can be used to find an orthogonal set.

Find a unitary matrix such that * is a diagonal matrix where

S O L U T I O N The eigenvectors of are shown after Example 5. Form the matrix by normalizing thesethree eigenvectors and using the results to create the columns of So, because

the unitary matrix is obtained.

Try computing the product * for the matrices and in Example 7 to see that youobtain

*

where and are the eigenvalues of A.�2�1, 6,

AP � ��1

0

0

0

6

0

0

0

�2�P

PAAPP

P � ��

17

1 � 2i7

17

1 � 2i7286 � 9i728

13728

1 � 3i40

�2 � i40

540

�P

�v3� � ��1 � 3i, �2 � i, 5�� 10 � 5 � 25 � 40,

�v2� � ��1 � 21i, 6 � 9i, 13�� 442 � 117 � 169 � 728

�v1� � ��1, 1 � 2i, 1�� 1 � 5 � 1 � 7

P.PA

A � �3

2 � i3i

2 � i0

1 � i

�3i1 � i

0�.

APPP

E X A M P L E 7 Diagonalization of a Hermitian Matrix

A

v2 � v3v1 � v3

� 0.

� �1 � 21i � 6 � 12i � 9i � 18 � 13

� ��1��1 � 21i� � �1 � 2i��6 � 9i� � 13

v1 � v2 � ��1��1 � 21i� � �1 � 2i��6 � 9i� � �1��13�

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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 is normal if it commutes with its conjugatetranspose: The main theorem of normal matrices states that a complex matrix

is normal if and only if it is unitarily diagonalizable. You are asked to explore normal matrices further in Exercise 65.

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 corre-spondence between unitary and Hermitian complex matrices when compared with orthogonal and symmetric real matrices.

AAA* � A*A.

A

A is a symmetric matrix A is a Hermitian matrix(real) (complex)

1. Eigenvalues of are real. 1. Eigenvalues of are real.2. Eigenvectors corresponding to 2. Eigenvectors corresponding to

distinct eigenvalues are orthogonal. distinct eigenvalues are orthogonal.3. There exists an orthogonal 3. There exists a unitary matrix

matrix such that such that

is diagonal. is diagonal.

P*APPTAP

PP

AA

Comparison of Symmetricand Hermitian Matrices

ExercisesSECTION 8.5In Exercises 1– 8, determine the conjugate transpose of the matrix.

1. 2.

3. 4.

5.

6.

7. 8.

In Exercises 9–12, use a graphing utility or computer software program to find the conjugate transpose of the matrix.

9.

10.

11.

12. A � �2 � i

0i

1 � 2i

12 � i2 � i

4

�12i

�i0

2i1 � i

1�2i

�A � �

1 � i2 � i1 � i

i

01i

2 � i

102

�1

�i2i4i0�

A � �i0

1 � 2i

11 � i

2i

1 � 2iii�

A � �1 � i

�2i

0i

2 � i

12 � i

2i�

A � �250

i3i

6 � i�A � �7 � 5i

2i

4�

A � �2 � i3 � i

3 � i2

4 � 5i6 � 2i�

A � �0

5 � i�2 i

5 � i64

2i43�

A � �4 � 3i2 � i

2 � i6i�A � �0

2

1

0�

A � �1 � 2i1

2 � i1�A � � i

2

�i

3i�

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In Exercises 13–16, explain why the matrix is not unitary.

13. 14.

15.

16.

In Exercises 17–22, determine whether is unitary by calculating*.

17. 18.

19. 20.

21. 22.

In Exercises 23–26, (a) verify that is unitary by showing that itsrows are orthonormal, and (b) determine the inverse of

23. 24.

25.

26.

In Exercises 27–34, determine whether the matrix is Hermitian.

27. 28.

29. 30.

31. 32.

33.

34.

In Exercises 35– 40, determine the eigenvalues of the matrix

35. 36.

37. 38.

39.

40.

In Exercises 41– 44, determine the eigenvectors of the matrix.

41. The matrix in Exercise 35




In Exercises 45–49, find a unitary matrix that diagonalizes thematrix

45. 46. A � � 02 � i

2 � i4�A � � 0

�ii0�

A.P

A � �100

4i0

1 � i3i

2 � i�

A � �2

i2

�i

2

�i

2

2

0

i2

0

2�

A � � 02 � i

2 � i4�A � � 3

1 � i1 � i

2�

A � � 3�i

i3�A � � 0

�ii0�

A.

A � �1

2 � i5

2 � i2

3 � i

53 � i

6�A � � 1

2 � i2 � i

23 � i3 � i�

A � �00

00�A � �1

001�

A � �0

2 � i0

ii1

100�A � �

02 � i

1

2 � ii0

101�

A � � i0

0�i�A � � 0

�ii0�

A

A � �0

�1 � i6

26

1

0

0

01 � i3

13

�A �

1

22 �3 � i

3 � i

1 � 3 i

1 � 3 i�

A � �1 � i

2

1 2

�1 � i

2

1 2

�A � ��45

3 5

3 5

4 5

i

i�A.

A

A � ��45

35

i

35

45

i�A � ��

i2

i2

0

i3

i3

i3

i6

i6

�i

6

�A � �

i2

i2

i2

�i

2�A � ��i

00i�

A � �1 � i1 � i

1 � i1 � i�A � �1 � i

1 � i1 � i1 � i�

AAA

A � �12

�i

3

�12

�12

13

12

1 � i2i

3

�1 � i

2�

A � �1 � i2

0

0

1

� i 2

0�A � �1

i

i

�1�A � � i

0

0

0�

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47.

48.

49.

50. Let be a complex number with modulus 1. Show that thematrix is unitary.

In Exercises 51–54, use the result of Exercise 50 to determine and such that is unitary.

51. 52.

53. 54.

In Exercises 55–58, prove the formula, where and are complex matrices.

55. 56.

57. 58.

59. Let be a matrix such that Prove that isHermitian.

60. Show that det where is a matrix.

In Exercises 61 and 62, assume that the result of Exercise 60 is truefor matrices of any size.

61. Show that

62. Prove that if is unitary, then

63. (a) Prove that every Hermitian matrix can be written as thesum where is a real symmetric matrix and

is real and skew-symmetric.

(b) Use part (a) to write the matrix

as the sum where is a real symmetric matrixand is real and skew-symmetric.

(c) Prove that every complex matrix can be written aswhere and are Hermitian.

(d) Use part (c) to write the complex matrix

as the sum where and are Hermitian.

64. Determine which of the sets listed below are subspaces of thevector space of complex matrices.

(a) The set of Hermitian matrices

(b) The set of unitary matrices

(c) The set of normal matrices

65. (a) Prove that every Hermitian matrix is normal.

(b) Prove that every unitary matrix is normal.

(c) Find a matrix that is Hermitian, but not unitary.

(d) Find a matrix that is unitary, but not Hermitian.

(e) Find a matrix that is normal, but neither Hermitiannor unitary.

(f) Find the eigenvalues and corresponding eigenvectors ofyour matrix from part (e).

(g) Show that the complex matrix

is not diagonalizable. Is this matrix normal?

66. Show that is unitary by computing


67. A complex matrix is called unitary if .

68. If is a complex matrix and is a complex number, then � kA*.�kA�*

kA

A�1 � A*A

AA*.A � In

� i0

1i�

2 � 2

2 � 2

2 � 2

n � n

n � n

n � n

n � n

CBA � B � iC,

A � � i2 � i

21 � 2i�

CBA � B � iC,An � n

CBA � B � iC,

A � � 21 � i

1 � i3�

CBA � B � iC,

A

�det�A�� 1.A

det�A*� � det�A�.

2 � 2A�A� � det�A�,

iAA* � A � O.A

�AB�* � B*A*�kA�* � kA*

�A � B�* � A* � B*�A*�* � A

n � nBA

A �12 �a

b

6 � 3i45

c �A �12 �

ib

ac�

A �12 �

3 � 4i5

b

a

c�A �

12 �

�1b

ac�

Aca, b,

A �12�

ziz

z�iz�

Az

A � ��1

00

0�1

�1 � i

0�1 � i

0�A � � 4

2 � 2i2 � 2i

6�

A � �2

i2

�i

2

�i

2

2

0

i2

0

2�

332600_08_5.qxp 4/17/08 11:31 AM Page 527


Review ExercisesCHAPTER 8In Exercises 1–6, perform the operation.

1. Find

2. Find

3. Find

4. Find

5. Find

6. Find

In Exercises 7–14, find all zeros of the polynomial function.

7. 8.

9. 10.

11. 12.

13.

14.

In Exercises 15–22, perform the operation using

and

15. 16. 17.

18. 19. 20.

21. 22.

In Exercises 23–28, perform the operation using and

23. 24. 25.

26. 27. 28.

In Exercises 29–32, perform the indicated operation.

29. 30.

31. 32.

In Exercises 33 and 34, find (if it exists).

33.

34.

In Exercises 35–40, determine the polar form of the complexnumber.

35. 36. 37.

38. 39. 40.

In Exercises 41–46, find the standard form of the complex number.

41.

42.

43. 44.

45. 46.

In Exercises 47–50, perform the indicated operation. Leave theresult in polar form.

47.

48.

49.

50.

In Exercises 51– 54, find the indicated power of the number and express the result in polar form.

51. 52.

53. 54.

In Exercises 55–58, express the roots in standard form.

55. Square roots:

56. Cube roots:

57. Cube roots: i

27cos

6� i sin

6�

25cos 2

3� i sin

2

3 �

�5cos

3� i sin

3��4

�2cos

6� i sin

6��7

�2i�3��1 � i�4

4 cos ��4� � i sin ��4��7 cos ��3� � i sin ��3��

9 cos��2� � i sin ��2��6 cos�2�3� � i sin �2�3��

�1

2 cos

2� i sin

2��2cos �

2� � i sin �

2��

�4cos

2� i sin

2��3cos

6� i sin

6��

4 cos � i sin �7cos 3

2� i sin

3

2 �

6cos 2

3� i sin

2

3 �4cos 5

4� i sin

5

4 �

2�cos�

3� � i sin�

3��

5�cos�

6� � i sin�

6��

3 � 2i7 � 4i1 � 3i

3 � i2 � 2i4 � 4i

A � �50

1 � ii�

A � � 3 � i�

235 �

115 i

�1 � 2i2 � 3i�A�1

5 � 2i

��2 � 2i��2 � 3i��1 � 2i��1 � 2i�

3 � 3i

1 � i

�1 � 2i

2 � i

2 � i

�zw�wv�vz��w�vz

z � �1 � 2i.v � 3 � i, w � 2 � 2i,

2AB3BA

det�A � B�det�A � B��iA

2iBA � BA � B

B � �1 � i2i

i2 � i�.A � �4 � i

32

3 � i�

p�x� � x4 � x3 � x2 � 3x � 6

p�x� � x4 � x3 � 3x2 � 5x � 10

p�x� � x3 � 2x � 4p�x� � x3 � 2x2 � 2x � 1

p�x� � x2 � 6x � 10p�x� � 3x2 � 3x � 3

p�x� � x2 � 4x � 7p�x� � x2 � 4x � 8

z � iu

z: u � 7 � i,

z � 3 � 3iu

z: u � 6 � 2i,

z � 1 � 2iuz: u � 2i,

z � 4 � 2iuz: u � 4 � 2i,

z � 8iu � z: u � 4,

z � 4iu � z: u � 2 � 4i,

332600_08_5.qxp 4/17/08 11:31 AM Page 528

Chapter 8 Rev iew E xerc ises 529

58. Fourth roots:

In Exercises 59–62, determine the conjugate transpose of thematrix.

59. 60.

61.

62.

In Exercises 63–66, find the indicated vector using

63. 64.

65. 66.

In Exercises 67 and 68, determine the Euclidean norm of the vector.

67. 68.

In Exercises 69 and 70, find the Euclidean distance between thevectors.

69.

70.

In Exercises 71–74, determine whether the matrix is unitary.

71. 72.

73. 74.

In Exercises 75 and 76, determine whether the matrix is Hermitian.

75.

76.

In Exercises 77 and 78, find the eigenvalues and correspondingeigenvectors of the matrix.

77. 78.

79. Prove that if is an invertible matrix, then * is also invertible.

80. Determine all complex numbers such that

81. Prove that if the product of two complex numbers is zero, thenone of the numbers must be zero.

82. (a) Find the determinant of the Hermitian matrix

(b) Prove that the determinant of any Hermitian matrix is real.

83. Let and be Hermitian matrices. Prove that if andonly if is Hermitian.

84. Let be a unit vector in Define Prove thatis an Hermitian and unitary matrix.

85. Use mathematical induction to prove DeMoivre’s Theorem.

86. Prove that if is a zero of a polynomial equation with real coefficients, then the conjugate of is also a zero.

87. Show that if and are both nonzero real numbers,then and are both real numbers.

88. Prove that if and are complex numbers, then

89. Prove that for all vectors and in a complex inner productspace,


90. A square complex matrix is called normal if it commutes withits conjugate transpose so that

91. A square complex matrix is called Hermitian if A � A*.A

AA* � A*A.A

� i�u � iv�2�.�u, v� �14 �u � v�2 � �u � v�2 � i�u � iv�2

vu

�z � w� ≤ �z� � �w�.wz

z2z1

z1z2z1 � z2

zz

n � nHH � I � 2uu*.Cn.u

ABAB � BABA

�3

2 � i3i

2 � i0

1 � i

�3i1 � i

0�.

z � �z.z

AA

�20i

030

�i02�� 4

2 � i2 � i

0�

�9

2 � i2

2 � i0

�1 � i

2�1 � i

3�

�1

1 � i2 � i

�1 � i3

�i

2 � ii4�

� 1 2

0

1 � i2

0

i

0

1 2

0

�1 � i2

��1i

0�i�

�2 � i

4

i 3

1 � i423

�� i 2 i 2

� 1 2 1 2

�

v � �2 � i, �1 � 2i, 3i�, u � �4 � 2i, 3 � 2i, 4�v � �2 � i, i�, u � �i, 2 � i�

v � �3i, �1 � 5i, 3 � 2i�v � �3 � 5i, 2i�

�3 � 2i�u � ��2i�wiu � iv � iw

3iw � �4 � i�v7u � v

v � �3, �i �, and w � �3 � i, 4 � i�.u � �4i, 2 � i�,

A � �2

�i1

1 � i2 � 2i1 � i

i0

�2i�

A � �5

2 � 2i3i

2 � i3 � 2i

2 � i

3 � 2ii

�1 � 2i�A � � 2 � i

1 � 2i2 � i

2 � 2i�A � ��1 � 4i3 � i

3 � i2 � i�

16cos

4� i sin

4�

332600_08_5.qxp 4/17/08 11:31 AM Page 529


ProjectsCHAPTER 8

1 Population Growth and Dynamical Systems – II

In the projects for Chapter 7, you were asked to model the populations of two speciesusing a system of differential equations of the form

The constants and depend on the particular species being studied. In Chapter7, you looked at an example of a predator-prey relationship, in which

Now consider a slightly different model.

1. Use the diagonalization technique to find the general solutions and at any time Although the eigenvalues and eigenvectors of the matrix

are complex, the same principles apply, and you can obtain complex exponential solutions.

2. Convert your complex solutions to real solutions by observing that ifis a (complex) eigenvalue of with (complex) eigenvector

then the real and imaginary parts of form a linearly independent pair of(real) solutions. You will need to use the formula

3. Use the initial conditions to find the explicit form of the (real) solutions tothe original equations.

4. If you have access to a computer software program or graphing utility, graphthe solutions obtained in part 3 over the domain At whatmoment are the two populations equal?

5. Interpret the solution in terms of the long-term population trend for the twospecies. Does one species ultimately disappear? Why or why not? Contrastthis solution to that obtained for the model in Chapter 7.

6. If you have access to a computer software program or graphing utility that can numerically solve differential equations, use it to graph the solutions tothe original system of equations. Does this numerical approximation appearto be accurate?

0 � t � 3.

ei� � cos � � i sin �.etv

v,A � a � bi

A � � 0.6�0.8

0.80.6�

t > 0.y2�t�y1�t�

y2�0� � 121y 2� �t� � �0.8y1�t� � 0.6y2�t�,

y1�0� � 36y 1� �t� � 0.6y1�t� � 0.8y2�t�,

c � �0.4, and d � 3.0.b � 0.6,a � 0.5,

da, b, c,

y 2� �t� � cy1�t� � dy2�t�.

y1� �t� � ay1�t� � by2�t�

332600_08_5.qxp 4/17/08 11:31 AM Page 530

complex vector spaces - cengage · 482 chapter 8 complex vector spaces complex numbers ... 9 12i 4...

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