8 Complex Vector Spaces - Cengage€¦ · 392 Chapter 8 Complex Vector Spaces 8.1 Complex Numbers...

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

8

391

Complex Vector Spaces

Quantum Mechanics (p. 425)

Signal Processing (p. 417)

Mandelbrot Set (p. 400)

Electric Circuits (p. 395)

Elliptic Curve Cryptography (p. 406)

Clockwise from top left, James Thew/Shutterstock.com; Mikhail/Shutterstock.com; Sybille Yates/Shutterstock.com; Lisa F. Young/Shutterstock; Stepán Kápl/Shutterstock.com

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392 Chapter 8 Complex Vector Spaces

8.1 Complex Numbers

Use the imaginary unit to write complex numbers.

Graphically represent complex numbers in the complex plane aspoints and as vectors.

Add and subtract two complex numbers, and multiply a complexnumber by a real scalar.

Multiply two complex numbers, and use the Quadratic Formula tofind all zeros of a quadratic polynomial.

Perform operations with complex matrices, and find the determinantof a complex matrix.

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 asThe general quadratic equation is and its

solutions are given by the Quadratic 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? Forinstance, the equation has a discriminant of but there isno real number whose square is To overcome this deficiency, mathematiciansinvented the imaginary unit defined as

where In terms of this imaginary unit,With this single addition of the imaginary unit to the real number system, the

system of complex numbers can be developed.

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. To see 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, twocomplex numbers are equal if and only if their real and imaginary parts are equal. Thatis, if and are two complex numbers written in standard form, then

if and only if and 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,

i��16 � 4��1 � 4i.i2 � �1.

i � ��1

i,�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.

i

REMARK

When working with productsinvolving square roots of negative numbers, be sure toconvert to a multiple of beforemultiplying. For instance, consider the following operations.

Correct

Incorrect � 1

� �1

��1��1 � ��1��1�

� �1

� i2

��1��1 � i � i

i

Definition of a Complex Number

If and are real numbers, then the number

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

bia

a � bi

ba

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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, complex numbers can be represented graphically as points in a coordinateplane 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 axisis the imaginary axis. The point that corresponds to the complex number is

as shown in Figure 8.1.

Plotting Numbers in the Complex Plane

Plot each number in the complex plane.

a. b. c. d. 5

SOLUTION

Figure 8.2 shows the numbers plotted in the complex plane.

a. b.

c. d.

Figure 8.2

Another way to represent the complex number is as a vector whose horizontal component is and whose vertical component is (See Figure 8.3.) (Notethat the use of the letter to represent the imaginary unit is unrelated to the use of torepresent a unit vector.)

Vector Representation of a Complex Number

Figure 8.3

1

−1

−2

−3

4 − 2iVertical

component

Horizontalcomponent

Realaxis

Imaginaryaxis

iib.a

a � bi

5 or (5, 0)Realaxis

Imaginaryaxis

−1 1 2 3 4 5

−2

1

2

3

4

−3i or (0, −3)

Realaxis

Imaginaryaxis

−1−2−3 1 2 3

−3

−2

−4

1

2

−2 − i

Realaxis

Imaginaryaxis

−1−2−3 1 2 3

−3

−4

1

2

or (−2, −1)

4 + 3ior (4, 3)

Realaxis

Imaginaryaxis

−1−2 1 2 3 4

−2

1

2

3

4

�3i�2 � i4 � 3i

�a, b�,a � bi

�a, b�.a � bi

8.1 Complex Numbers 393

(a, b) or a + bi

Realaxis

Imaginaryaxis

The Complex Plane

a

b

Figure 8.1

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ADDITION, SUBTRACTION, AND SCALAR

MULTIPLICATION OF COMPLEX NUMBERS

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

Adding and Subtracting Complex Numbers

a.

b.

Using the vector representation of complex numbers, you can add or subtract two complex numbers geometrically using the parallelogram rule for vector addition, asshown in Figure 8.4.

Figure 8.4

Many of the properties of addition of real numbers are valid for complex numbersas 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 ofaddition in the natural way. For example,

� 3 � 3i.

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

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

� �2 � 4i

�1 � 3i� � �3 � i� � �1 � 3� � ��3 � 1�i � 5

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

i,

Definition of Addition and Subtraction of Complex Numbers

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 � bi

REMARK

Note in part (a) of Example 2that the sum of two complexnumbers can be a real number.

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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.

Scalar Multiplication with Complex Numbers

a.

b.

Geometrically, multiplication of a complex number by a real scalar corresponds tothe multiplication of a vector by a scalar, as shown in Figure 8.5.

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 toverify this in Exercise 55.

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


LINEAR ALGEBRA APPLIED

Complex numbers have some useful applications in electronics. The state of a circuit element is described bytwo quantities: the voltage across it and the current flowing through it. To simplify computations, the circuit element’s state can be described by a single complex number of which the voltage and current aresimply the real and imaginary parts. A similar notation canbe used to express the circuit element’s capacitance andinductance.

When certain elements of a circuit are changing with time, electrical engineers often have to solve differentialequations. These can often be simpler to solve using complex numbers because the equations are less complicated.

z � V � li,

IV

Definition of Scalar Multiplication

If is a real number and is a complex number, then the scalar multipleof and is defined as

c�a � bi� � ca � cbi.

a � bica � bic

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MULTIPLICATION OF COMPLEX NUMBERS

The operations of addition, subtraction, and scalar multiplication of complex numbershave exact counterparts with the corresponding vector operations. By contrast, there isno direct vector counterpart for the multiplication of two complex numbers.

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

Distributive property


Use

Commutative property


Multiplying Complex Numbers

a.

b.

Complex Zeros of a Polynomial

Use the Quadratic Formula to find the zeros of the polynomial and verify that for each zero.

SOLUTION

Using the Quadratic Formula,

Substitute each value of into the polynomial to verify that

In Example 5, the two complex numbers and are complex conjugates of each other (together they form a conjugate pair). More will be saidabout complex conjugates in Section 8.2.

3 � 2i3 � 2i

� 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

� 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� � 0.p�x�x

�6 � ��16

2�

6 � 4i

2� 3 � 2i. x �

�b � �b2 � 4ac

2a

p�x� � 0p�x� � x2 � 6x � 13

� 11 � 2i

� 8 � 3 � 6i � 4i

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

��2��1 � 3i� � �2 � 6i

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

i2 � �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�TECHNOLOGY

Many graphing utilities andsoftware programs can calculate with complex numbers. For example, onsome graphing utilities, youcan express a complex number

as an ordered pair Try verifying the result ofExample 4(b) by multiplying

and You shouldobtain the ordered pair �11, 2�.

�4, 3�.�2, �1�

�a, b�.a � bi

Definition of Multiplication of Complex Numbers

The product of the complex numbers and is defined as

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

c � dia � bi

REMARK

A well-known result from algebra states that the complexzeros of a polynomial with realcoefficients must occur in conjugate pairs. (See ReviewExercise 81.)

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COMPLEX MATRICES

Now that you are able to add, subtract, and multiply complex numbers, you can applythese operations to matrices whose entries are complex numbers. Such a matrix iscalled complex.

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

Operations with Complex Matrices

Let and be the complex matrices

and

and determine each of the following.

a. b. c. d.

SOLUTION

a.

b.

c.

d.

Finding the Determinant of a Complex Matrix

Find the determinant of the matrix

SOLUTION

� �8 � 26i

� 10 � 20i � 6i � 12 � 6

� �2 � 4i��5 � 3i� � �2��3�

det�A� � �2 � 4i3

25 � 3i�

A � �2 � 4i3

25 � 3i�.

� � �27 � i

�2 � 2i3 � 9i�

� � �2 � 0�1 � 2 � 3i � 4i � 6

2i � 2 � 0i � 1 � 4 � 8i�

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


Definition of a Complex Matrix

A matrix whose entries are complex numbers is called a complex matrix.

TECHNOLOGY

Many graphing utilities andsoftware programs can performmatrix operations on complexmatrices. Try verifying the calculation of the determinantof the matrix in Example 7. You should obtain the sameanswer, ��8, �26�.

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8.1 Exercises

Simplifying an Expression In Exercises 1–6, determinethe value of the expression.

1. 2. 3.

4. 5. 6.

Equality of Complex Numbers In Exercises 7–10,determine such that the complex numbers in each pairare equal.

7.

8.

9.

10.

Plotting Complex Numbers In Exercises 11–16, plotthe number in the complex plane.

11. 12. 13.

14. 15. 16.

Adding and Subtracting Complex Numbers InExercises 17–24, find the sum or difference of the complex numbers. Use vectors to illustrate your answer.

17. 18.

19. 20.

21. 22.

23. 24.

Scalar Multiplication In Exercises 25 and 26, use vectors to illustrate the operations geometrically. Be sureto graph the original vector.

25. and , where

26. and , where

Multiplying Complex Numbers In Exercises 27–34,find the product.

27. 28.

29. 30.

31. 32.

33. 34.

Finding Zeros In Exercises 35–40, determine all thezeros of the polynomial function.

35. 36.

37. 38.

39. 40.

Finding Zeros In Exercises 41–44, use the given zeroto find all zeros of the polynomial function.

41. Zero:

42. Zero:

43. Zero:

44. Zero:

Operations with Complex Matrices In Exercises45–54, perform the indicated matrix operation using thecomplex matrices and

and

45. 46.

47. 48.

49. 50.

51. det 52. det

53. 54.

55. Proof Prove that the set of complex numbers, with theoperations of addition and scalar multiplication (withreal scalars), is a vector space of dimension 2.

57. (a) Evaluate for and 5.

(b) Calculate

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

58. Let

(a) Calculate for and 5.

(b) Calculate

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

True or False? In Exercises 59 and 60, determinewhether each statement is true or false. If a statement istrue, give a reason or cite an appropriate statement fromthe text. If a statement is false, provide an example thatshows the statement is not true in all cases or cite anappropriate statement from the text.

59. 60.

61. Proof Prove that if the product of two complex numbersis zero, then at least one of the numbers must be zero.

��10�2 � �100 � 10��2��2 � �4 � 2

n.An

A2010.

n � 1, 2, 3, 4,An

A � �0i

i0�.

n.in

i2010.

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 � 10x2 � 9p�x� � x4 � 16

p�x� � x2 � 4x � 5p�x� � x2 � 5x � 6

p�x� � x2 � x � 1p�x� � 2x2 � 2x � 5

�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�

u � 2 � i�32u3u

u � 3 � i2u�u

�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�

z � 1 � 5iz � 1 � 5iz � 7

z � �5 � 5iz � 3iz � 6 � 2i

��x � 4� � �x � 1�i, x � 3i

�x2 � 6� � �2x�i, 15 � 6i

�2x � 8� � �x � 1�i, 2 � 4i

x � 3i, 6 � 3i

x

��i�7i 4i 3

��4��4�8��8��2��3

56. Consider the functions and

(a) Without graphing either function, determinewhether the graphs of and have -intercepts.Explain your reasoning.

(b) For which of the given functions is azero? Without using the Quadratic Formula, find theother zero of this function and verify your answer.

x � 3 � i

xqp

q�x� � x2 � 6x � 10.p�x� � x2 � 6x � 10

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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 zerosof 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.

Finding the Conjugate of a Complex Number

Complex Number Conjugate

a.

b.

c.

d.

Geometrically, two points in the complex plane are conjugates if and only if theyare reflections in the real (horizontal) axis, as shown in Figure 8.6. Complex conjugateshave many useful properties. Some of these are shown in Theorem 8.1.

PROOF

To prove the first property, let Then and

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

Finding the Product of Complex Conjugates

When you have zz � �1 � 2i��1 � 2i� � 12 � 22 � 1 � 4 � 5.z � 1 � 2i,

� a � bi � a � bi � z.� �a � bi��z�

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

z � a � biz � a � bi.

z � 5z � 5

z � 2iz � �2i

z � 4 � 5iz � 4 � 5i

z � �2 � 3iz � �2 � 3i

3 � 2i.3 � 2ip�x� � x2 � 6x � 13

Definition of the Conjugate of a Complex Number

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

z � a � bi.

zz � a � bi

REMARK

In part (d) of Example 1, notethat 5 is its own complex conjugate. In general, it can be shown that a number is itsown complex conjugate if andonly if the number is real. (SeeExercise 39.)

Conjugate of a Complex Number

Figure 8.6

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

THEOREM 8.1 Properties of Complex Conjugates

For a complex number the following properties are true.

1. 2.3. if and only if 4. �z� � zz � 0.zz � 0

zz � 0zz � a2 � b2

z � a � bi,

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THE MODULUS OF A COMPLEX NUMBER

Because a complex number can be represented by a vector in the complex plane, itmakes sense to talk about the length of a complex number. This length is called themodulus of the complex number.

Finding the Modulus of a Complex Number

For and determine the value of each modulus.

a. b. c.

SOLUTION

a.

b.

c. Because you have

Note that in Example 3, In Exercise 41, you are asked to prove that this multiplicative property of the modulus always holds. Theorem 8.2 states that themodulus of a complex number is related to its conjugate.

PROOF

Let then and 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

Definition of the Modulus of a Complex Number

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

�z� � �a2 � b2.

�z�z � a � bi


Fractals appear in almost every part of the universe. Theyhave been used to study a wide variety of applications suchas bacteria cultures, the human lungs, the economy, andgalaxies. The most famous fractal is called the MandelbrotSet, named after the Polish-born mathematician BenoitMandelbrot (1924–2010). The Mandelbrot Set is based onthe following sequence of complex numbers.

The behavior of this sequence depends on the value of thecomplex number For some values of the modulus ofeach term in the sequence is less than some fixed number

and the sequence is bounded. This means that is in theMandelbrot Set, and its point is colored black. For other values of the moduli of the terms of the sequence becomeinfinitely large, and the sequence is unbounded. This meansthat is not in the Mandelbrot Set, and its point is assigned a color based on “how quickly” the sequence diverges.

c

c,

cN,zn

c,c.

zn � �zn�1�2 � c, z1 � c

REMARK

The modulus of a complexnumber is also called theabsolute value of the number.In fact, when is a real number,

�z� � �a2 � 02 � �a�.z

THEOREM 8.2 The Modulus of a Complex Number

For a complex number �z�2 � zz.z,

Andrew Park/Shutterstock.com

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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 performingdivision in the complex number system. To define division of complex numbers,consider and and assume that and are not both 0. For the quotient

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 following definition is obtained.

In practice, the quotient of two complex numbers can be found by multiplying thenumerator and the denominator by the conjugate of the denominator, as follows.

Division of Complex Numbers

a.

b.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

�ac � bd

c2 � d2�

bc � ad

c2 � d2i

��ac � bd� � �bc � ad�i

c2 � d 2

��a � bi��c � di��c � di��c � di�

a � bi

c � di�

a � bi

c � di �c � di

c � di�

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

Definition of Division of Complex Numbers

The quotient of the complex numbers and is defined as

provided c2 � d2 � 0.

�1

�w�2 �zw �

�ac � bd

c2 � d2�

bc � ad

c2 � d2i

z

w�

a � bi

c � di

w � c � diz � a � bi

REMARK

If then and In other words, asis the case with real numbers,division of complex numbersby zero is not defined.

w � 0.c � d � 0,c2 � d2 � 0,

9781133110873_0802.qxp 3/10/12 6:57 AM Page 401


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

and verify your solution by showing that

SOLUTION

Using the formula for the inverse of a matrix from Section 2.3,

Furthermore, because

it follows that

To verify your solution, multiply and as follows.

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

PROOF

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

� �10

01�

110�

100

010 AA�1 � �2 � i

3 � 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

TECHNOLOGY

If your graphing utility or software program can performoperations with complex matrices, then you can verifythe result of Example 5. If youhave matrix stored on agraphing utility, evaluate A�1.

A

THEOREM 8.3 Properties of Complex Conjugates

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,z

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8.2 Exercises 403

8.2 Exercises

40. Consider the quotient

(a) Without performing any calculations, describehow to find the quotient.

(b) Explain why the process described in part (a)results in a complex number of the form

(c) Find the quotient.

a � bi.

1 � i6 � 2i

.

Finding the Conjugate In Exercises 1–6, find the complex conjugate and geometrically represent both

and

1. 2.

3. 4.

5. 6.

Finding the Modulus In Exercises 7–12, find the indicated modulus, where and

7. 8.

9. 10.

11. 12.

13. Verify that where and

14. Verify that where and

Dividing Complex Numbers In Exercises 15–20,perform the indicated operations.

15. 16.

17. 18.

19. 20.

Operations with Complex Rational Expressions InExercises 21–24, perform the operation and write theresult in standard form.

21. 22.

23. 24.

Finding Zeros In Exercises 25–28, use the given zero tofind all zeros of the polynomial function.

25. Zero:

26. Zero:

27. Zero:

28. Zero:

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

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

29. 30.

Finding the Inverse of a Complex Matrix In Exercises31–36, determine whether the complex matrix has aninverse. If is invertible, find its inverse and verify that

31. 32.

33. 34.

35. 36.

Singular Matrices In Exercises 37 and 38, determineall values of the complex number for which is singular. (Hint: Set and solve for )

37. 38.

39. Proof Prove that if and only if is real.

41. Proof Prove that for any two complex numbers andeach of the statements below is true.

(a)

(b) If then

42. Graphical Interpretation Describe the set of points in the complex plane that satisfies each of thestatements below.

(a) (b)

(c) (d)

43. (a) Evaluate for 1, 2, 3, 4, and 5.

(b) Calculate and

(c) Find a general formula for for any positiveinteger

44. (a) Verify that

(b) Find the two square roots of

(c) Find all zeros of the polynomial x4 � 1.

i.

�1 � i�2 �2

� i.

n.�1i�n

�1i�2010.�1i�2000

n ��1i�n

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

2i1 � i

0

1 � iz0A � � 5

3iz

2 � i

z.det�A� � 0Az

A � �100

01 � i

0

00

1 � iA � �i00

0i0

00i

A � �1 � i0

21 � iA � �1 � i

12

1 � i

A � �2i3

�2 � i3iA � � 6

2 � i3ii

AA�1 � I.A

A

z � 1 � iz � 2 � i

z�2z�1z3z2

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

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

9781133110873_0802.qxp 3/10/12 6:57 AM Page 403


8.3 Polar Form and DeMoivre’s Theorem

Determine the polar form of a complex number, convert betweenthe 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 complexnumbers in polar form.

POLAR FORM OF A COMPLEX NUMBER

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 suchas When you use the four basic operations (addition, subtraction, multiplication, anddivision), there seems to be no reason to guess that

That is,

To work effectively with powers and roots of complex numbers, it is helpful to use apolar representation for complex numbers, as shown in Figure 8.7. Specifically, if

is a nonzero complex number, then let be the angle from the positive real axisto the radial line passing through the point and let be the modulus of This leads to the following.

So, from which the polar form of a complex number is obtained.

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 thatsatisfies the inequality

Principal argument

is called the principal argument and is denoted by Arg( ). Two nonzero complex numbers in polar form are equal if and only if they have the same modulus and the same principal argument.

z

�� < � � �

��

a � bi � �r cos �� r sin ��i,

r � �a2 � b2

b � r sin �

a � r cos �

a � bi.r�a, b��a � bi

�1 � i�2 �2

� i.

�i �1 � i�2

.

i.

Definition of the Polar Form of a Complex Number

The polar form of the nonzero complex number is given by

where and The number is the modulus 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 � bi

REMARK

The polar form of is expressed as

where is any angle.�

z � 0�cos � � i sin ��,

z � 0

Imaginaryaxis

Realaxis

br

a 0

θ

θ θ Standard Form: a + bi

Polar Form: r(cos + i sin )

(a, b)

Figure 8.7

9781133110873_0803.qxp 3/10/12 6:54 AM Page 404

Finding the Polar Form of a Complex Number

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

a. b. c.

SOLUTION

a. Because and which implies thatFrom and

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 8.8.

a. b.

c.

Figure 8.8

z � 1�cos �

2� i sin

�

2�

Imaginaryaxis

Realaxis

θ

z = i1

1

z � �13 cos�0.98� � i sin�0.98�z � �2 �cos��

4� � i sin��

4��

Realaxis1 2

4

3

2

1θ

z = 2 + 3i

Imaginaryaxis

Imaginaryaxis

Realaxis

−1−1−2

1

2

2

−2

θ

z = 1 − i

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

z � iz � 2 � 3iz � 1 � i

8.3 Polar Form and DeMoivre’s Theorem 405

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Converting from Polar to Standard Form

Express the complex number in standard form.

SOLUTION

Because and 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 75.)

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

z � 8�cos��

3� � i sin��

3��


Elliptic curves are the foundation for elliptic curve cryptography (ECC), a type of public key cryptography forsecure communications over the Internet. ECC has gainedpopularity due to its computational and bandwidth advantages over traditional public key algorithms.

One specific variety of elliptic curve is formed usingEisenstein integers. Eisenstein integers are complex

numbers of the form where

and and are integers. These numbers can be graphed as intersection points of a triangular lattice in the complexplane. Dividing the complex plane by the lattice of allEisenstein integers results in an elliptic curve.

ba

� � �12

��32

i,z � a � b�,

THEOREM 8.4 Product and Quotient of Two Complex Numbers

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�

thumb/Shutterstock.com

9781133110873_0803.qxp 3/10/12 6:54 AM Page 406

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 subtractarguments. (See Figure 8.9.)

Figure 8.9

Multiplying and Dividing in Polar Form

Find and for the complex numbers

and

SOLUTION

Because and are in polar form, apply Theorem 8.4, as follows.

multiply

add adddivide

subtract subtract

Use the standard forms of and to check the multiplication in Example 3. For instance,

To verify that this answer is equivalent to the result in Example 3, use the formulas forand to obtain

and sin�5�

12� � sin��

6�

�

4� ��6 ��2

4.cos�5�

12� � cos��

6�

�

4� ��6 ��2

4

sin �u � v�cos �u � v�

�53 �

�6 � �24

��6 � �2

4i�.

�5�6 � 5�2

12�

5�6 � 5�212

i

z1z2 � �5�22

�5�2

2i��3

6�

16

i� �5�612

�5�212

i �5�612

i �5�212

i2

z2z1

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�

z2z1

z2 �13�cos

�

6� i sin

�

6�.z1 � 5�cos �

4� i sin

�

4�z1�z2z1z2

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.


REMARK

Try verifying the division inExample 3 using the standardforms of and z2.z1

9781133110873_0803.qxp 3/10/12 6:54 AM Page 407

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

and

Similarly,

and

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.

Raising a Complex Number

to an Integer Power

Find and write the result in standard form.

SOLUTION

First convert to polar form. For

and

which implies that So,

By DeMoivre’s Theorem,

� 4096.

� 40961 � i�0� � 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

z5 � r 5�cos 5� � i sin 5��.

z4 � r4�cos 4� � i sin 4��

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

� r2�cos 2� � i sin 2�� z2 � r �cos � � i sin ��r �cos � � i sin �� z � r �cos � � i sin ��


THEOREM 8.5 DeMoivre’s Theorem

If and is any positive integer, then

zn � rn�cos n� � i sin n��.

nz � r�cos � � i sin ��

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Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomial of degree has zeros in the complex number system. So, a polynomialsuch as has six zeros, and in this case you can find the six zeros by factoring and using 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 seehow this is done, let be an th root of where

and

Then, by DeMoivre’s Theorem

and because it follows that

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

Finally, substituting this value of into the polar form of produces the result statedin the next 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 sin ��.w � s�cos � i sin �

z,nw

n

x �1 ± �3 i

2.x �

�1 ± �3i2

,x � ±1,

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

p�x� � x6 � 1nn


REMARK

Note that when exceedsthe roots begin to

repeat. For instance, whenthe angle is

which yields the same valuesfor the sine and cosine ask � 0.

� � 2�nn

��

n� 2�

k � n,

n � 1,k

Definition of the nth Root of a Complex Number

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

z � wn � �a � bi�n.

znw � a � bi

THEOREM 8.6 The nth Roots 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,

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The formula for the th roots of a complexnumber has a nice geometric interpretation, asshown in Figure 8.10. Because the th roots allhave the same modulus (length) they lie on acircle of radius with center at the origin.Furthermore, the roots are equally spacedaround the circle, because successive th rootshave arguments that differ by

You have already found the sixth roots of 1 byfactoring and the Quadratic Formula. Try solvingthe same problem using Theorem 8.6 to get theroots shown in Figure 8.11. When Theorem 8.6 isapplied to the real number 1, the th roots have aspecial name—the th roots of unity.

Finding the nth Roots of a Complex Number

Determine the fourth roots of

SOLUTION

In polar form,

so Then, by applying Theorem 8.6,

Setting and 3,

as shown in Figure 8.12.

Figure 8.12

Imaginaryaxis

Realaxis

13π

π8

π8

9π8

813π

8

9π8cos + i sin

5π8

5π8cos + i sin

cos + i sin

cos + i sin

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,

� 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.

nn

2��n.n

n

n�r

n�r,n

n


rn

2π

2π

n

n

The nth Roots of aComplex Number

Realaxis

Imaginaryaxis

Figure 8.10

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

−

−

Figure 8.11

REMARK

In Figure 8.12, note that wheneach of the four angles

and ismultiplied by 4, the result is ofthe form ��2� � 2k�.

13��8��8, 5��8, 9��8,

9781133110873_0803.qxp 3/10/12 6:54 AM Page 410

8.3 Exercises 411

8.3 Exercises

Converting to Polar Form In Exercises 1–4, expressthe complex number in polar form.

1. 2.

3. 4.

Graphing and Converting to Polar Form In Exercises5–16, represent the complex number graphically, andgive the polar form of the number. (Use the principalargument.)

5. 6.

7. 8.

9. 10.

11. 7 12. 4

13. 14.

15. 16.

Graphing and Converting to Standard Form InExercises 17–26, represent the complex number graphically, and give the standard form of the number.

17.

18.

19.

20.

21. 22.

23. 24.

25. 26.

Multiplying and Dividing in Polar Form In Exercises27–34, perform the indicated operation and leave theresult in polar form.

27.

28.

29.

30.

31.

32.

33.

34.

Finding Powers of Complex Numbers In Exercises35–44, use DeMoivre’s Theorem to find the indicatedpowers of the complex number. Express the result instandard form.

35. 36.

37. 38.

39.

40.

41.

42.

43.

44.

Finding Square Roots of a Complex Number InExercises 45–52, find the square roots of the complexnumber.

45. 46.

47. 48.

49. 50.

51. 52. 1 � �3 i1 � �3 i

2 � 2i2 � 2i

�6i�3i

5i2i

�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

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

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

cos�5��3� � i sin�5��3�cos� � i sin�

2cos�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��

9�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�15

4 �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

Imaginaryaxis

Realaxis

3i

1

1−1

2

3

Imaginaryaxis

Realaxis

−6 1

2

−2

−2−3−4−5−6

−3

3

Imaginaryaxis

Realaxis

1 + 3i

1 2

1

2

3

−1

Imaginaryaxis

Realaxis

1

−1

−2

2

2 − 2i

9781133110873_0803.qxp 3/10/12 6:54 AM Page 411


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) expresseach of the roots in standard form.

53. Square roots:

54. Square roots:

55. Fourth roots:

56. Fifth roots:

57. Square roots:

58. Fourth roots:

59. Cube roots:

60. Cube roots:

61. Cube roots: 8

62. Fourth roots:

63. Fourth roots: 1

64. Cube roots: 1000

Finding and Graphing Solutions In Exercises 65–72,find all the solutions of the equation and represent yoursolutions graphically.

65. 66.

67. 68.

69. 70.

71. 72.

73. Electrical Engineering In an electric circuit, the formula relates voltage drop current andimpedance where complex numbers can representeach of these quantities. Find the impedance when thevoltage drop is and the current is

75. Proof When provided with two complex numbersand

with prove that

76. Proof Show that the complex conjugate ofis

77. Use the polar forms of and in Exercise 76 to findeach of the following.

(a)

(b)

78. Proof Show that the negative of is

79. Writing

(a) Let

Sketch and in the complex plane.

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

80. 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 beexpressed in polar form as

(c) Prove that if then

(d) Prove the formula


81. Although the square of the complex number is givenby the absolute value of the complex number is defined as

82. Geometrically, the th roots of any complex number are all equally spaced around the unit circle centered atthe 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!�

x4

4!�

x6

6!� . . . .

sin x � x �x 3

3!�

x5

5!�

x7

7!� . . .

ex � 1 � x �x 2

2!�

x3

3!�

x4

4!� . . .

xx,ex,

i?zi?z

z�iiz,z,

z � r�cos � � i sin �� 2�cos �

6� i sin

�

6�.

�z � rcos�� i sin�� .

z � r�cos � � i sin ��z�z, z 0

zz

zz

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

z1z2

�r1r2

cos��1 � �2� � i sin��1 � �2�.

z2 0,z2 � r2�cos �2 � i sin �2�,z1 � r1�cos �1 � i sin �1�

2 � 4i.5 � 5i

Z,I,V,V � I � Z

x4 � i � 0x 3 � 64i � 0

x4 � 81 � 0x5 � 243 � 0

x3 � 27 � 0x 3 � 1 � 0

x4 � 16i � 0x4 � 256i � 0

81i

�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�

74. Use the graph of the roots of acomplex number.

(a) Write each of the roots in trigonometric form.

(b) Identify the complex number whose roots aregiven. Use a graphing utility to verify your results.

(i) (ii)

Realaxis

Imaginaryaxis

45°45°

45°45°

3 3

3 3

30°30°

−11

Realaxis

Imaginaryaxis

2 2

2

9781133110873_0803.qxp 3/10/12 6:54 AM Page 412

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 spacesrepresent a vector in by a basis, and find the Euclidean

inner product, Euclidean norm, and Euclidean distance in

Recognize complex inner product spaces.

COMPLEX VECTOR SPACES

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

where the scalars are complex numbers. The complex version of is the complex vector space consisting of ordered -tuples of complex numbers. So,a vector in 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 performedcomponent by component.

Vector Operations in

Let

and

be vectors in the complex vector space Determine each vector.

a.

b.

c.

SOLUTION

a. In column matrix form, the sum is

b. Because and

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� � �5i, 7 � i �.

�2 � i�v � �2 � i��1 � 2i, 3 � 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�

Cn

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

Cn.CnCn,

9781133110873_0804.qxp 3/10/12 6:56 AM Page 413

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

which is the standard basis for Because this basis contains vectors, it follows thatthe dimension of is Other bases exist; in fact, any linearly independent set of vectors in can be used, as demonstrated in Example 2.

Verifying a Basis

Show that is a basis for

SOLUTION

Because the dimension of is 3, the set will be a basis if it is linearly independent. To check for linear independence, set a linear combination of the vectorsin equal to as follows.

This implies that

So, and is linearly independent.

Representing a Vector in by a Basis

Use the basis in Example 2 to represent the vector

SOLUTION

By writing

you can obtain

which implies that

and

So, 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

v � �2, i, 2 � i�.S

Cn

�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�C3

C 3.S � �v1, v2, v3� � ��i, 0, 0�, �i, i, 0�, �0, 0, i��

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


REMARK

Try verifying that this linearcombination yields �2, i, 2 � i �.

9781133110873_0804.qxp 3/10/12 6:56 AM Page 414

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

The Space of Complex-Valued 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 numbersforms the scalars for and vector addition is defined by

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

be a complex number. Then

is in

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

Finding the Euclidean Inner Product in

Determine the Euclidean inner product of the vectors

and

SOLUTION

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

C3

Rn,Cn

S.

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

c � a � biS

S,

� �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

m � nCn,


REMARK

Note that if and happen tobe “real,” then this definitionagrees with the standard inner(or dot) product in Rn.

vu

THEOREM 8.7 Properties of the Euclidean Inner Product

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 � 0u � u � 0

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

kCnwu, v,

Definition of the Euclidean Inner Product in

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

u � v � u1v1 � u2v2 � � � � � unvn.

vuCn.vu

Cn

9781133110873_0804.qxp 3/10/12 6:56 AM Page 415


Definitions of the Euclidean Norm and Distance in

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�1�2.

uCnu

Cn

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

and

Then

The Euclidean inner product in is used to define the Euclidean norm (or length)of a vector in and the Euclidean distance between two vectors in .

The Euclidean norm and distance may be expressed in terms of components, asfollows (see Exercise 51).

Finding the Euclidean Norm and Distance in

Let and

a. Find the norms of and b. Find the distance between and

SOLUTION

a.

b.

� �47

� �1 � 5 � 41�1�2

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

� �1, �2 � i, 4 � 5i� d�u, v� � u � v

� �7

� �2 � 5 � 0�1�2

� ��12 � 12� � �22 � 12� � �02 � 02�1�2

v � � v1 2 � v2 2 � v3 2�1�2

� �46

� �5 � 0 � 41�1�2

� ��22 � 12� � �02 � 02� � �42 � ��5�2�1�2

u � � u1 2 � u2 2 � u3 2�1�2

v.uv.u

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

Cn

d�u, v� � � u1 � v1 2 � u2 � v2 2 � . . . � un � vn 2�1�2

u � � u1 2 � u2 2 � . . . � un 2�1�2

CnCnCn

� u � v.

� u1v1 � u2v2 � � � � � unvn

� v1u1 � v2u2 � . . . � vnun

� v1u1 � v2u2 � . . . � vnun

v � u � v1u1 � v2u2 � . . . � vnun

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

9781133110873_0804.qxp 3/10/12 6:56 AM Page 416

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 aninner product, use the properties listed in Theorem 8.7.

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 and be vectors in the complex space Show that thefunction defined by

is a complex inner product.

SOLUTION

Verify the four properties of a complex inner product, as follows.

1.

2.

3.

4.

Moreover, if and only if

Because all the 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�

Cn


Definition of a Complex Inner Product

Let and be vectors in a complex vector space. A function that associates and with the complex number is called a complex inner product whenit satisfies the following properties.

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�vuvu


Complex vector spaces and inner products have an importantapplication called the Fourier transform, which decomposes a function into a sum of orthogonal basis functions. The givenfunction is projected onto the standard basis functions for varying frequencies to get the Fourier amplitudes foreach frequency. Like Fourier coefficients and the Fourier approximation, this transform is named after the Frenchmathematician 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 thetwo notes, and yet your brain is able to separate the notes.The Fourier transform gives a mathematical way to take asignal and separate out its frequency components.

Eliks/Shutterstock.com

9781133110873_0804.qxp 3/10/12 6:56 AM Page 417


8.4 Exercises

Vector Operations In Exercises 1–8, perform the indicated operation using

and

1. 2.

3. 4.

5. 6.

7. 8.

Linear Dependence or Independence In Exercises9–12, determine whether the set of vectors is linearlyindependent or linearly dependent.

9.

10.

11.

12.

Verifying a Basis In Exercises 13–16, determinewhether is a basis for

13.

14.

15.

16.

Representing a Vector by a Basis In Exercises 17–20,express as a linear combination of each of the followingbasis vectors.

(a)

(b)

17. 18.

19. 20.

Finding Euclidean Inner Products In Exercises 21 and22, determine the Euclidean inner product

21. 22.

Properties of Euclidean Inner Products In Exercises23–26, let and Evaluate the expressions in parts (a) and (b)to verify that they are equal.

23. (a) 24. (a)

(b) (b)

25. (a) 26. (a)

(b) (b)

Finding the Euclidean Norm In Exercises 27–34,determine the Euclidean norm of

27. 28.

29. 30.

31. 32.

33.

34.

Finding the Euclidean Distance In Exercises 35–40,determine the Euclidean distance between and

35.

36.

37.

38.

39.

40.

Complex Inner Products In Exercises 41–44, determinewhether the function is a complex inner product, where

and

41.

42.

43.

44.

Finding Complex Inner Products In Exercises 45–48,use the inner product to find

45. and

46. and

47. and

48. and

Finding Complex Inner Products In Exercises 49 and50, use the inner product

where

and

to find

49.

50. v � � i3i

�2i�1�u � � 1

1 � i2i0�

v � �10

1 � 2ii�u � �0

1i

�2i��u, v�.

v � [v11

v21

v12

v22]u � [u11

u21

u12

u22]

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

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�

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.

k�u � v�k�u � v�u � �kv��ku� � v

u � w � v � wv � u

�u � v� � wu � v

k � �i.w � �1 � i, 0�,v � �2i, 2 � i�,u � �1 � i, 3i�,

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

u � 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

��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��

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�,

9781133110873_0804.qxp 3/10/12 6:56 AM Page 418

8.4 Exercises 419

51. Let

(a) Use the definitions of Euclidean norm and Euclideaninner product to show that

(b) Use the results of part (a) to show that

53. Let and If and the set is not a basis for what doesthis imply about and

54. Let and Determine a vectorsuch that is a basis for

Properties of Complex Inner Products In Exercises55–58, verify the statement using the properties of acomplex inner product.

55.

56.

57.

58.

Proof In Exercises 59–63, prove the property, whereand are vectors in and is a complex number.

59.

60.

61.

62.

63. if and only if

64. Writing Let be a complex inner product and letbe a complex number. How are and

related?

Finding a Linear Transformation In Exercises 65 and66, determine the linear transformation thathas the given characteristics.

65.

66.

Finding an Image and a Preimage In Exercises 67–70,the linear transformation is shown by

Find the image of and the preimage of

67.

68.

69.

70.

71. Find the kernel of the linear transformation in Exercise 68.

72. Find the kernel of the linear transformation in Exercise 69.

Finding an Image In Exercises 73 and 74, find theimage of for the indicated composition, where

and are the matrices below.

and

73. 74.

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

(a) The set of symmetric matrices.

(b) The set of matrices satisfying

(c) The set of matrices in which all entries arereal.

(d) The set of diagonal matrices.

76. Determine which of the sets below are subspaces of the vector space of complex-valued functions (seeExample 4).

(a) The set of all functions satisfying

(b) The set of all functions satisfying

(c) The set of all functions satisfying


77. Using the Euclidean inner product of and in

78. The Euclidean norm of in denoted by is�u � u�2.

uCnu

u � v � u1v1 � u2v2 � . . . � unvn.Cn,vu

f�i� � f��i�.f

f�0� � 1.f

f�i� � 0.f

2 � 2

2 � 2

�A�T � A.A2 � 2

2 � 2

2 � 2

T1 � T2T2 � 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 � �0i

i0

10�,

w � �00�v � �1 � i

1 � i�,A � �1i

0i�,

w.vT�v� � Av.T : Cm → Cn

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

T : Cm → Cn

�u, kv��u, v�k�u, v�

u � 0.u � u � 0

u � u � 0

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

kCnwu, v,

� k�u, v��u, kv��u, v� � �u, v� � �v, 2u��u, 0� � 0

�u, kv � w� � k�u, v� � �u, w�

C3.�v1, v2, v3�v3

v2 � �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�

� . . . � un � vn 2�1�2.

d�u, v� � � u1 � v1 2 � u2 � v2 2

u � � u1 2 � u2 2 � . . . � un 2�1�2.

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

52. The complex Euclidean innerproduct of and is sometimes called the complex dot product. Compare the properties ofthe complex dot product in and those of the dotproduct in

(a) Which properties are the same? Which propertiesare different?

(b) Explain the reasons for the differences.

Rn.Cn

vu

9781133110873_0804.qxp 3/10/12 6:56 AM Page 419


8.5 Unitary and Hermitian Matrices

Find the conjugate transpose of a complex matrix

Determine if a matrix is unitary.

Find the eigenvalues and eigenvectors of a Hermitian matrix, anddiagonalize a Hermitian matrix.

CONJUGATE TRANSPOSE OF A MATRIX

Problems involving diagonalization of complex matrices and the associated eigenvalueproblems require the concepts of unitary and Hermitian matrices. These matricesroughly correspond to orthogonal and symmetric real matrices. In order to define unitary and Hermitian matrices, the concept of the conjugate transpose of a complexmatrix must first be introduced.

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

Finding the Conjugate Transpose

of a Complex Matrix

Determine * for the matrix

SOLUTION

Several properties of the conjugate transpose of a matrix are listed in the followingtheorem. The proofs of these properties are straightforward and are left for you to supply in Exercises 47–50.

A* � AT � �3 � 7i0

�2i4 � i�

A � �3 � 7i2i

04 � i� � �3 � 7i

�2i0

4 � i�

A � �3 � 7i2i

04 � i�.

A

� ATAA

A

A.A*

Definition of the Conjugate Transpose of a Complex Matrix

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

*

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

A

� A TA

AA,

THEOREM 8.8 Properties of the Conjugate Transpose

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

1. 2.3. 4. �AB�* � B*A*�kA�* � kA*

�A � B�* � A* � B*�A*�* � A

kBA

9781133110873_0805.qxp 3/10/12 6:55 AM Page 420

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 matricesare called unitary.

A Unitary Matrix

Show that the matrix is unitary.

SOLUTION

Begin by finding the product *.

Because

it follows that So, 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

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

1.

2.

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

i � jvi � vj � 0,

�vi� � 1, i � 1, 2, . . . , m

Cn

�v1, v2, . . . , vm�

AA* � A�1.

AA* � �10

01� � I2

� �10

01�

�14�

40

04�

AA* �12�

1 � i1 � i

1 � i1 � i�

12 �

1 � i1 � i

1 � i1 � i�

AA

A �12 �

1 � i1 � i

1 � i1 � i�A

A�1 � AA�1 � AT.A

8.5 Unitary and Hermitian Matrices 421

Definition of Unitary Matrix

A complex matrix is unitary when

A�1 � A*.

A

THEOREM 8.9 Unitary Matrices

An complex matrix is unitary if and only if its row (or column) vectorsform an orthonormal set in Cn.

An � n

9781133110873_0805.qxp 3/10/12 6:55 AM Page 421

The Row Vectors of a Unitary Matrix

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

SOLUTION

Let and be defined as follows.

Begin by showing that and are unit vectors.

Then show that all pairs of distinct vectors are orthogonal.

So, is an orthonormal set.�r1, r2, r3�

� 0

��5

65�

1 � 3i

65�

4 � 3i

65

r2 � r3 � �i

3�5i

215� � i3�

3 � i

215� � 13�

4 � 3i

215 � � 0

� �5i

415�

4 � 2i

415�

�4 � 3i

415

r1 � r3 � 12�

5i

215� � 1 � i2 � 3 � i

215� � �12�

4 � 3i

215 � � 0

�i

23�

i

23�

1

23�

1

23

r1 � r2 � 12��

i3� � 1 � i

2 � i3� � �

12�

13�

� 1

� �2560

�1060

�2560�

1�2

�r3� � � 5i

215�5i

215� � 3 � i

215�3 � i

215� � 4 � 3i

215 �4 � 3i

215 ��1�2

� 1

� �13

�13

�13�

1�2

�r2� � �� i3��

i3� � i

3�i

3� � 13�

13��

1�2

�r1� � �12�

12� � 1 � i

2 �1 � i2 � � �

12��

12��

1�2

� �14

�24

�14�

1�2� 1

r3r2,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


REMARK

Try showing that the columnvectors of also form anorthonormal set in C3.

A

9781133110873_0805.qxp 3/10/12 6:55 AM Page 422

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 conjugatetranspose. Such a matrix is called Hermitian after the French mathematician CharlesHermite (1822–1901).

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

The conjugate transpose of has the form

If is Hermitian, then So, must be of the form

Similar results can be obtained for Hermitian matrices of order In other words,a square 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 theentry in the th row and the th column.

Hermitian Matrices

Which matrices are Hermitian?

a. b.

c. d.

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 andsecond 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.

��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 �

ijaji

jiaij

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


Definition of a Hermitian Matrix

A square matrix is Hermitian when

A � A*.

A

9781133110873_0805.qxp 3/10/12 6:55 AM Page 423

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

PROOF

Let be an eigenvalue of and let

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

Furthermore, because

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

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

Finding the Eigenvalues of a Hermitian Matrix

Find the eigenvalues of the matrix

SOLUTION

The characteristic polynomial of is

So, the characteristic equation is and the eigenvalues of are and �2.�1, 6,

A�� 1�� 6�� 2� � 0,

� �� 1�� 6�� 2�. � �3 � 3�2 � 16� � 12

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

� �� 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.

�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�


THEOREM 8.10 The Eigenvalues of a Hermitian Matrix

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

REMARK

Note that this theorem impliesthat the eigenvalues of a realsymmetric matrix are real, asstated in Theorem 7.7.

9781133110873_0805.qxp 3/10/12 6:55 AM Page 424

To find the eigenvectors of a complex matrix, use a procedure similar to that usedfor a real matrix. For instance, in Example 5, to find the eigenvector corresponding tothe eigenvalue substitute the value for into the equation

to obtain

Solve this equation using Gauss-Jordan elimination, or a graphing utility or softwareprogram, to 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 � �2�2 � 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,



Quantum mechanics had its start in the early 20th centuryas scientists began to study subatomic particles and light.Collecting data on energy levels of atoms, and the rates oftransition between levels, they found that atoms could beinduced to more excited states by the absorption of light.

German physicist Werner Heisenburg (1901–1976) laid amathematical foundation for quantum mechanics usingmatrices. Studying the dispersion of light, he used vectors torepresent energy levels of states and Hermitian matrices torepresent “observables” such as momentum, position, andacceleration. He noticed that a measurement yields preciselyone real value and leaves the system in precisely one of a setof mutually exclusive (orthogonal) states. So, the eigenvaluesare the possible values that can result from a measurementof an observable, and the eigenvectors are the correspondingstates of the system following the measurement.

Let matrix be a diagonal Hermitian matrix that representsan observable. Then consider a physical system whosestate is represented by the column vector To measure thevalue of the observable in the system of state you canfind the product

Because is Hermitian and its values along the diagonalare real, is a real number. It represents the average of the values given by measuring the observable on asystem in the state a large number of times.u

Au*Au

A

� �u1u1�a11 � �u2u2�a22 � �u3u3�a33.

u*Au � u1 u2 u3��a11

00

0a22

0

00

a33��

u1

u2

u3�

u,Au.

A

TECHNOLOGY

Some graphing utilities andsoftware programs have built-inprograms for finding the eigenvalues and correspondingeigenvectors of complex matrices.

Jezper/Shutterstock.com

9781133110873_0805.qxp 3/10/12 6:55 AM Page 425


Just as real symmetric matrices are orthogonally diagonalizable, Hermitian matrices are unitarily diagonalizable. A square matrix is unitarily diagonalizablewhen there exists a unitary matrix such that

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

PROOF

To prove part 1, let and be two eigenvectors corresponding to the distinct (andreal) eigenvalues and Because and you have the equations shown below 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 a Hermitian Matrix

The eigenvectors of the Hermitian matrix shown in Example 5 are mutually orthogonalbecause the eigenvalues are distinct. Verify this by calculating the Euclidean inner products and For example,

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

The three eigenvectors in Example 6 are mutually orthogonal because they correspond to distinct eigenvalues of the Hermitian matrix . Two or more eigenvectorscorresponding to the same eigenvalue may not be orthogonal. Once any set of linearlyindependent eigenvectors is obtained for an eigenvalue, however, the Gram-Schmidtorthonormalization process can be used to find an orthogonal set.

A

v2 � v3v1 � v3

� 0.

� �1 � 21i � 6 � 9i � 12i � 18 � 13

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

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

v2 � v3.v1 � v3,v1 � v2,

v2v1

�1� �2 v1*v2 � 0

��2 � �1�v1*v2 � 0

�2v1*v2 � �1v1*v2 � 0

v2�1v2 � �1v1v2 � v1v2 � ��1v1��Av1�v2�2v2 � �2v1Av2 � v1v2 � v1v2 � v1�Av1�

v2.�Av1�Av2 � �2v2,Av1 � �1v1�2.�1

v2v1

APPPAP�1 � PP

P�1AP

PA

THEOREM 8.11 Hermitian Matrices and Diagonalization

If is an Hermitian matrix, then

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

n � nA

9781133110873_0805.qxp 3/10/12 6:55 AM Page 426

Diagonalization of a Hermitian Matrix

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

SOLUTION

The eigenvectors of are shown after Example 5. Form the matrix by normalizingthese three eigenvectors and using the results to create the columns of

So,

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

*

where and are the eigenvalues of 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 unitarilydiagonalizable. A square complex matrix is normal when it commutes with its conjugate transpose: The main theorem of normal matrices states that a complex matrix is normal if and only if it is unitarily diagonalizable. You are askedto explore normal matrices further in Exercise 56.

The properties of complex matrices described in this section are comparable to theproperties of real matrices discussed in Chapter 7. The summary below indicates thecorrespondence between unitary and Hermitian complex matrices when compared withorthogonal and symmetric real matrices.

AAA* � A*A.

A

A.�2�1, 6,

AP � ��1

0

0

0

6

0

0

0

�2�P

PAAPP

P � ��

17

1 � 2i7

17

1 � 21i7286 � 9i728

13728

1 � 3i40

�2 � i40

540

� .

�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


Comparison of Symmetric and Hermitian Matrices

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 2. Eigenvectors corresponding

to distinct eigenvalues are to distinct eigenvalues are orthogonal. 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

9781133110873_0805.qxp 3/10/12 6:55 AM Page 427


8.5 Exercises

Finding the Conjugate Transpose In Exercises 1–4,determine the conjugate transpose of the matrix.

1. 2.

3. 4.

Finding the Conjugate Transpose In Exercises 5 and6, use a software program or graphing utility to find theconjugate transpose of the matrix.

5.

6.

Non-Unitary Matrices In Exercises 7–10, explain whythe matrix is not unitary.

7. 8.

9.

10.

Identifying Unitary Matrices In Exercises 11–16,determine whether is unitary by calculating *.

11. 12.

13. 14.

15.

16.

Row Vectors of a Unitary Matrix In Exercises 17–20,(a) verify that is unitary by showing that its rows areorthonormal, and (b) determine the inverse of

17. 18.

19.

20.

Identifying Hermitian Matrices In Exercises 21–26,determine whether the matrix is Hermitian.

21. 22.

23. 24.

25.

26.

Finding Eigenvalues of a Hermitian Matrix InExercises 27–32, determine the eigenvalues of the matrix

27. 28.

29. 30.

31. A � �100

4i0

1 � i3i

2 � i�A � � 0

2 � i2 � i

4�A � � 31 � i

1 � i2�

A � � 3�i

i3�A � � 0

�ii0�

A.

�1

2 � i5

2 � i2

3 � i

53 � i

6�� 1

2 � i2 � i

23 � i3 � i�

�0

2 � i0

ii1

100��

02 � i

1

2 � ii0

101�

� i0

0�i�� 0

�ii0�

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 � ��

i2

i2

0

i3

i3

i3

i6

i6

�i

6

�

A � ��45

35

i

35

45

i�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�

�2 � i

0i

1 � 2i

12 � i2 � i

4

�12i

�i0

2i1 � i

1�2i

��1 � i2 � i1 � i

i

01i

2 � i

102

�1

�i2i4i0�

�250

i3i

6 � i��0

5 � i�2 i

5 � i64

2i43�

�1 � 2i1

2 � i1�� i

2

�i

3i�

9781133110873_0805.qxp 3/10/12 6:55 AM Page 428

8.5 Exercises 429

32.

Finding Eigenvectors of a Hermitian Matrix InExercises 33–36, determine the eigenvectors of thematrix in the indicated exercise.

33. Exercise 27 34. Exercise 30

35. Exercise 31 36. Exercise 28

Diagonalization of a Hermitian Matrix In Exercises37–41, find a unitary matrix that diagonalizes thematrix

37. 38.

39.

40.

41.

43. Show that is unitary by computing

44. Let be a complex number with modulus 1. Show thatthe matrix is unitary.

Unitary Matrices In Exercises 45 and 46, use the result ofExercise 44 to determine and such that is unitary.

45. 46.

Proof In Exercises 47–50, prove the formula, where and are complex matrices.

47. 48.

49. 50.

51. Proof Let be a matrix such that Provethat is Hermitian.

52. Show that det where is a matrix.

Determinants In Exercises 53 and 54, assume that theresult of Exercise 52 is true for matrices of any size.

53. Show that

54. Prove that if is unitary, then

55. (a) Prove that every Hermitian matrix can be written asthe sum where is a real symmetricmatrix and is real and skew-symmetric.

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

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

(c) Prove that every complex matrix can bewritten as where and areHermitian.

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

as the sum where and areHermitian.

56. (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 notHermitian.

(e) Find a matrix that is normal, but neitherHermitian nor unitary.

(f) Find the eigenvalues and corresponding eigenvectorsof your matrix in part (e).

(g) Show that the complex matrix

is not diagonalizable. Is this matrix normal?

� i0

1i�

2 � 2

2 � 2

2 � 2

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�,iA

A* � 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 �

�1b

ac�

Aca, b,

A �12�

ziz

z�iz�

Az

AA*.A � In

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�

A � � 02 � i

2 � i4�A � � 0

�ii0�

A.P

A � �2

i2

�i

2

�i

2

2

0

i2

0

2�

42. Consider the following matrix.

(a) Is unitary? Explain.

(b) Is Hermitian? Explain.

(c) Are the row vectors of orthonormal? Explain.

(d) The eigenvalues of are distinct. Is it possible to determine the inner products of the pairs ofeigenvectors by inspection? If so, state thevalue(s). If not, explain why not.

(e) Is unitarily diagonalizable? Explain.A

A

A

A

A

A � ��2

3 � i4 � i

3 � i1

1 � i

4 � i1 � i

3�

9781133110873_0805.qxp 3/10/12 6:55 AM Page 429


Operations with Complex Numbers In Exercises1–6, perform the operation.

1. Find

2. Find

3. Find

4. Find

5. Find

6. Find

Finding Zeros In Exercises 7–10, use the given zero tofind all zeros of the polynomial function.

7. Zero:

8. Zero:

9. Zero:

10. Zero:

Operations with Complex Matrices In Exercises11–18, perform the operation using

and

11. 12.

13. 14.

15. 16.

17. 18.

Operations with Conjugates and Moduli In Exercises19–24, perform the operation using

and

19. 20. 21.

22. 23. 24.

Dividing Complex Numbers In Exercises 25–28,perform the indicated operation.

25. 26.

27. 28.

Finding the Inverse of a Complex Matrix In Exercises29 and 30, find (if it exists).

29.

30.

Converting to Polar Form In Exercises 31–36,determine the polar form of the complex number.

31. 32.

33. 34.

35. 36.

Converting to Standard Form In Exercises 37–42,find the standard form of the complex number.

37.

38.

39. 40.

41. 42.

Multiplying and Dividing in Polar Form In Exercises43–46, perform the indicated operation. Leave the resultin polar form.

43.

44.

45.

46.

Finding Powers of Complex Numbers In Exercises47–50, find the indicated power of the number andexpress the result in polar form.

47. 48.

49. 50.

Finding Roots of Complex Numbers In Exercises51–54, express the roots in standard form.

51. Square roots:

52. Cube roots:

53. Cube roots:

54. Fourth roots: 16�cos �

4� i sin

�

4�i

27�cos �

6� i sin

�

6�

25�cos 2�

3� i sin

2�

3 �

�5�cos �

3� i sin

�

3��4

��2�cos �

6� i sin

�

6��7

�2i�3��1 � i�4

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

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

�1

2 �cos �

2� i sin

�

2��2�cos ��

2� � i sin ��

2��4�cos

�

2� i sin

�

2��3�cos �

6� i sin

�

6��

4cos � � i sin ��7�cos 3�

2� i sin

3�

2 �

6�cos 2�

3� i sin

2�

3 �4�cos 5�

4� i sin

5�

4 �

2�cos��

3� � i sin��

3��

5�cos��

6� � i sin��

6��

3 � 2i7 � 4i

1 � �3i�3 � i

2 � 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��iA2iB

A � BA � B

B � �1 � i2i

i2 � i�.A � �4 � i

32

3 � i�

�3ip�x� � x4 � x3 � x2 � 3x � 6

��5ip�x� � x4 � x3 � 3x2 � 5x � 10

�1 � ip�x� � x3 � 2x � 4

�2ip�x� � x3 � 2x2 � 2x � 4

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,

8 Review Exercises

9781133110873_08REV.qxp 3/10/12 6:53 AM Page 430

Review Exercises 431

Vector Operations in In Exercises 55–58, find theindicated vector using and

55. 56.

57. 58.

Finding the Euclidean Norm In Exercises 59 and 60,determine the Euclidean norm of the vector.

59. 60.

Finding the Euclidean Distance In Exercises 61 and62, find the Euclidean distance between the vectors.

61.

62.

Finding Complex Inner Products In Exercises 63–66,use the inner product to find

63. and

64. and

65. and

66. and

Finding the Conjugate Transpose In Exercises 67–70,determine the conjugate transpose of the matrix.

67.

68.

69.

70.

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

71. 72.

73. 74.

Identifying Hermitian Matrices In Exercises 75 and76, determine whether the matrix is Hermitian.

75.

76.

Finding Eigenvalues and Eigenvectors In Exercises77 and 78, find the eigenvalues and corresponding eigenvectors of the matrix.

77. 78.

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

80. Determine all complex numbers such that

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

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

(b) Proof Prove that the determinant of anyHermitian matrix is real.

83. Proof Let and be Hermitian matrices. Prove thatif and only if is Hermitian.

84. Proof Let be a unit vector in DefineProve that is an Hermitian and

unitary matrix.

85. Proof Use mathematical induction to proveDeMoivre’s Theorem.

86. Proof Show that if and are both nonzeroreal numbers, then and are both real numbers.

87. Proof Prove that if and are complex numbers,then

88. Proof Prove that for all vectors and in a complexinner product space,


89. A square complex matrix is called normal if it commutes with its conjugate transpose so that

90. A square complex matrix is called Hermitian ifA � 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

n � nHH � I � 2uu*.Cn.u

ABAB � BABA

�3

2 � i3i

2 � i0

1 � i

�3i1 � i

0�.

zz

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 � i4�2�3

�� i �2 i �2

� 1 �2 1 �2

�

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�

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

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

�u, v�.�u, v� � u1v1 � 2u2v2

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

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

Cn

9781133110873_08REV.qxp 3/10/12 6:53 AM Page 431

8 Project

1 Population Growth and Dynamical Systems (II)

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

The constants and depend on the particular species being studied. InChapter 7, 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 atany time Although the eigenvalues and eigenvectors of the matrix

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

2. Convert the complex solutions to real solutions by observing that if is a (complex) eigenvalue of with (complex) eigenvector then the real andimaginary parts of form a linearly independent pair of (real) solutions. Usethe formula

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

4. If you have access to a graphing utility or software program, graph the solutionsobtained in part 3 over the domain At what moment 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? Contrast thissolution to that obtained for the model in Chapter 7.

6. If you have access to a graphing utility or software program that can numericallysolve differential equations, use it to graph the solutions of the original systemof equations. Does this numerical approximation appear to 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�


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8 Complex Vector Spaces - Cengage€¦ · 392 Chapter 8 Complex Vector Spaces 8.1 Complex Numbers...

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