lineal capitulo 8

7/21/2019 Lineal Capitulo 8

1/49

8 Complex Vector Spaces8.1 Complex Numbers8.2 Conjugates and Division

of Complex Numbers

8.3 Polar Form and

DeMoivres Theorem

8.4 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 DeMoivres 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*

Cn.

Cn.

Cn

Cn.

Cn

.

481

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482 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s

Complex Numbers

So far in the text, the scalar quantities used have been real numbers. In this chapter, you

will expand the set of scalars to include complex numbers.

In algebra it is often necessary to solve quadratic equations such as

The general quadratic equation is and its solutions are given by the

Quadratic Formula,

where the quantity under the radical, is called the discriminant. If

then the solutions are ordinary real numbers. But what can you conclude

about the solutions of a quadratic equation whose discriminant is negative? For example, the

equation has a discriminant of From your experience with

ordinary 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 is

To 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 following

operations.

Correct

Incorrect

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

of complex numbers can be developed.

i

11 11 1 1

11 i i i2 1

i

i

16 41 4i.

i2 1.

i,1.

16 161 161 41,

16.

b2 4ac 16.x2 4 0

b2 4ac 0,

b2 4ac,

xbb2 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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Se ct io n 8.1 Com ple x Nu mb ers 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. To

see this, note that every real number can be written as a complex number usingThat is, for every real number,

A complex number is uniquely determined by its real and imaginary parts. So, you can

say that two complex numbers are equal if and only if their real and imaginary parts are

equal. 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 the

complex plane. This plane is an adaptation of the rectangular coordinate plane.

Specifically, the horizontal axis is the real axis and the vertical axis is the imaginary axis.

For instance, Figure 8.1 shows the graph of two complex numbers, and The

number is associated with the point and the number is associated with

the point

Another way to represent the complex number is as a vector whose horizontal

component 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.)

ii

b.a

a bi2,1.

2 i3, 23 2i2 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 the

number. The form is the standard form of a complex number.a bi

bia

a bi

baDefinition of aComplex Number

(3, 2) or 3 + 2i

Realaxis

Imaginaryaxis

11

1

2

3

2 32

(2, 1)

or 2 iThe Complex Plane

Figure 8.1

y1

x1

1

1

2

3

4 2iVertical

component

Vector Representation of aComplex Number

Horizontal

component

Figure 8.2

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/ /


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Addition and Scalar Multiplication of Complex NumbersBecause 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 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 shown

in Figure 8.3.

Figure 8.3

Realaxis

1

1

2

3

3 2 3

Imaginary

axis

w= 3 + i

z= 1 3i

zw= 2 4i

Subtraction of Complex Numbers

1

2

3

4

1 2 3 41 65

2

34

Addition of Complex Numbersw= 2 4i

z+ w= 5

z= 3 + 4i

Imaginary

axis

Real

axis

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

2 4i 3 4i 2 3 4 4i 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

Differencea bi c di a c b di

a bi c di a c b di

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 as

well. For instance, addition of complex numbers is both associative and commutative.

Moreover, to find the sum of three or more complex numbers, extend the definition of

addition in the natural way. For example,

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 the

multiplication 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 space

of dimension 2 (where the scalars are the real numbers). You are asked to verify this in

Exercise 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

Imaginary

axis

1 6i

41 i 23 i 31 4i 4 4i 6 2i 3 12i

38 17i

32 7i 48 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 4i

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

is defined as

ca bi ca cbi.

a bi

ca bicDefinition of

Scalar Multiplication

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Multiplication of Complex NumbersThe operations of addition, subtraction, and multiplication by a real number have

exact counterparts with the corresponding vector operations. By contrast, there is no direct

counterpart for the multiplication of two complex numbers.

Rather than try to memorize this definition of the product of two complex numbers, you

should 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.px 0

px 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 312 i4 3i 8 6i 4i 3i221 3i 2 6i

E X A M P L E 3 Multiplying Complex Numbers

ac bd ad bci

ac bd adi bci

i 2 1. ac adi bci bd1

ac adi bci bdi2a bic di ac di bic 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, 3.2,1a,b.a bi

Te chnologyNote

The product of the complex numbers and is defined as

a bic di ac bd ad bci .

c dia biDefinition of

Multiplication of

Complex Numbers

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SOLUT ION 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 of

each other (together they form a conjugate pair). A well-known result from algebra states

that the complex zeros of a polynomial with real coefficients must occur in conjugate pairs.

(See Review Exercise 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 these

operations 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, as

demonstrated in the next two examples.

3 2i3 2i

9 12i 4 18 12i 13 0.

9 6i 6i 4 18 12i 13

3 2i3 2i 63 2i 13

p3 2i 3 2i2 63 2i 13

9 12i 4 18 12i 13 0

9 6i 6i 4 18 12i 13

3 2i3 2i 63 2i 13

p3 2i 3 2i2 63 2i 13

pxx

6 16

2

6 4i

2 3 2i.

xbb2 4ac

2a

6 36 52

2

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

Complex Matrix

_ _ q p / / g

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Let and be the complex matrices below

and

and determine each of the following.

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

SOLUT ION (a)

(b)

(c)

(d)

Find the determinant of the matrix

SOLUT ION

8 26i

10 20i 6i 12 6

detA

2 4i

32

5 3i 2 4i5 3i 23

A 2 4i32

5 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 0

i 1 4 8i

2

7 i

2 2i

3 9i

BA 2ii0

1 2i i

2 3i

1 i

4

A B i2 3i1 i

4 2i

i

0

1 2i 3i

2 2i

1 i

5 2i

2 iB 2 i2ii0

1 2i 2 4i

1 2i

0

4 3i

3A 3

i

2 3i

1 i

4

3i

6 9i

3 3i

12

BAA B2 iB3A

B 2ii0

1 2iA i

2 3i

1 i

4BA

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

Many computer software programs and graphing utilities are capable of performing matrixoperations on complex matrices. Try verifying the calculation of the determinant of the matrix in

Example 6. You should obtain the same answer, 8,26.

Te chnology

Note

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ExercisesSECTION 8.1In Exercises 16, determine the value of the expression.

1. 2.

3. 4.

5. 6.

In Exercises 712, plot the complex number.

7. 8.

9. 10.11. 12.

In Exercises 13 and 14, use vectors to illustrate the operations

graphically. Be sure to graph the original vector.

13. and , where

14. and , where

In Exercises 1518, determine such that the complex numbers in

each pair are equal.

15.

16.

17.

18.

In Exercises 1926, 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 2736, find the product.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

In Exercises 3742, determine the zeros of the polynomial

function.

37.

38.

39.

40.

41.

42.

In Exercises 4346, use the given zero to find all zeros of the

polynomial function.

43. Zero:

44. Zero:

45. Zero:

46. Zero:

In Exercises 4756, perform the indicated matrix operation using

the 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 of

addition and scalar multiplication (with real scalars), is a vector

space 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.AnA2010.

n 1, 2, 3, 4,An

A 0ii

0.n.in

i 2010.

n 1, 2, 3, 4,in

BA5AB

BA B

14iB2iA

12B2A

B AA B

B 1 i3

3i

iA 1 i

2 2i

1

3iB.A

x 3ipx x3 x2 9x 9

x 5ipx 2x3 3x2 50x 75

x 4px x3 2x2 11x 52

x 1px x3 3x2 4x 2

px x4 10x2 9

px x4 16

px x2 4x 5

px x2 5x 6

px x2 x 1

px 2x2 2x 5

1 i21 i2a bi3

2 i2 2i4 i1 i3

a bia bia bi2

4 2 i4 2 i7 i7 i

3 i 23 i5 5i1 3i

2 i 2 i2 i 2 i12 7i 3 4i6 2i

i 3 i5 i 5 i

1 2 i 2 2 i2 6i 3 3i

x 4 x 1i,x 3i

x2 6 2xi, 15 6i

2x 8 x 1i, 2 4i

x 3i, 6 3i

x

u 2 i32u3u

u 3 i2uu

z 1 5iz 1 5iz 7z 5 5i

z 3iz 6 2i

i7i 4i 344

8823

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60. Prove that if the product of two complex numbers is zero, then

at least one of the numbers must be zero.

True or False? In Exercises 61 and 62, determine whether each

statement is true or false. If a statement is true, give a reason or cite

an appropriate statement from the text. If a statement is false,

provide an example that shows the statement is not true in all cases

or cite an appropriate statement from the text.

61.

62. 102 100 1022 4 2

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Se c ti on 8 .2 Co nju gat es and Di v is io n of Co mp le x Numb ers 491

Conjugates and Division of Complex NumbersIn 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 by

changing 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 number

is 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 5

z 2iz 2i

z 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 2ipxx2 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 aComplex 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.

PROOF To prove the first property, let Then and

The second and third properties follow directly from the first. Finally, the fourth property

follows the definition of the complex conjugate. That is,

Find the product of and its complex conjugate.

SOLUT ION Because you have

zz

1

2i1

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 biz

zz a bia bi a2 abi abi b2i2 a2 b2.

z a biz a bi.

Realaxis

2 32

234

5

3 5 6 7

1

2

3

4

5

z= 4 5i

z= 4 + 5i

Imaginary

axis

Realaxis

z= 2 3i

z= 2 + 3i

2

2

3

34 1 1 2

3

Imaginary

axis

For a complex number the following properties are true.

1.

2.

3. if and only if

4. z zz 0.zz 0

zz 0

zz 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 makes

sense to talk about the length of a complex number. This length is called the modulus of

the complex number.

R E M A R K : The modulus of a complex number is also called the absolute value of the

number. In fact, when is a real number, you have

For and determine the value of each modulus.

(a) (b) (c)SOLUT ION (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 modulus

of a complex number is related to its conjugate.

PROOF Let then and you have

zz a bia bi a2 b2 z2.

z a biz a bi,

zw zw.

zw 152 162 481.

zw 2 3i6 i 15 16i,

w 62 12 37

z 22 32 13

zwwz

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.

zz a biDefinition of the

Modulus of a

Complex Number

For a complex number

z2zz.

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 is

obtained.

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 adi

c2

d2

a bi

c di

a bi

c dic dic di

a bic di

c dic di

w 0.c d 0,c2 d2 0,

zw a bic di ac bd bc adi,

y bc ad

ww.x

ac bd

ww

yx

dx cy b

cx dy a

z a bi,

z wxyi c dixyi cx dy dx cyi.

z

wxyi

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

w2zw

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 of

a complex matrix, as demonstrated in Example 5.

Find the inverse of the matrix

and verify your solution by showing that

SOLUT ION 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.

100

11

1010

0

0

10AA

1

2 i

3 i

5 2i

6 2i1

1020

10

17 i

7 i

A1A

1

1020

10

17 i

7 i.

1

3 i1

3 i6 2i3 i

3 i3 i

5 2i3 i

2 i3 i

A1 13 i6 2i3 i 5 2i2 i

3 i

12 6i 4i 2 15 6i 5i 2

A 2 i6 2i 5 2i3 i

A1 1

A 6 2i3 i

5 2i2 i.

2 2

AA1 I2.

A 2 i3 i5 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 4i3 4i

3 4i 2 11i

9 16

2

25

11

25i

1

1 i

1

1 i1 i1 i

1 i

12 i2

1 i

2

1

2

1

2i

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

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ExercisesSECTION 8.2

In Exercises 16, find the complex conjugate and graphically

represent both and

1. 2.

3. 4.

5. 6.

In Exercises 712, find the indicated modulus, where

and

7. 8. 9.

10. 11. 12.

13. Verify that where and

14. Verify that where and

In Exercises 1520, perform the indicated operations.

15. 16.

17. 18.

19. 20.3 i

2 i5 2i

2 i3 i

4 2i

5 i

4 i

3 2 i

3 2 i

1

6 3i

2 i

i

v 2 3i.

z 1 2izv2

zv2

zv2,

w 1 2i.

z 1 iwz wz zw,

zv2

vwz

zwz2

z

v 5i.w 3 2i,

z 2 i,

z 3z 4

z 2iz 8i

z 2 5iz 6 3i

z.z

z


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

conjugates.

PROOF To prove the first property, let and Then

The proof of the second property is similar. The proofs of the other two properties are

left to you.

z w.

a bi c di

a c b di

z w a c b di

w c di.z a bi

If your computer software program or graphing utility can perform operations with complexmatrices, then you can verify the result of Example 5. If you have matrix stored on a graphingutility, evaluateA1.

ATe chnology

Note

For the complex numbers and the following properties are true.

1.2.

3.

4. zw zw

zw zw

z w z wz w z w

w,zTHEOREM 8.3

Properties ofComplex Conjugates

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In Exercises 2124, perform the operation and write the result in

standard form.

21. 22.

23. 24.

In Exercises 2528, 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 3136, determine whether the complex matrix has an

inverse. 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 number

for 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 the

statements below is true.

(a)

(b) If then

41. Describe the set of points in the complex plane that satisfies

each of the statements below.

(a)

(b)

(c)

(d)


statement is true or false. If a statement is true, give a reason or

cite an appropriate statement from the text. If a statement is false,



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 satisfies

each 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 polynomialx4 1.

i.

1 i22

i.

n.1in1i2010.1i2000

1in

z > 3z 1 1z i 2z 4

i i 2 0

2

z

5

z i 2z 1 i 5z 3

zw zw.w 0,zw zw

w,z

zz z

A 2

1 i

1

2i

1 i

0

1 i

z

0

A 53iz

2 iz.detA 0Az

A i

0

0

0

i

0

0

0

iA

1

0

0

0

1 i

0

0

0

1 i

A 1 i02

1 iA 1 i

1

2

1 i

A 2i32 i

3iA 6

2 i

3i

iAA

1

I.A

A

z 1 i

z 2 i

z2z1z 3z 2

z.

1 3ipx x3 4x2 14x 20

3 2 ipx x4 3x3 5x2 21x 22

3

ipx

4x3

23x2

34x

10

1 3 ipx 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


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Polar Form and DeMoivres TheoremAt this point you can add, subtract, multiply, and divide complex numbers. However, there

is still one basic procedure that is missing from the algebra of complex numbers. To see

this, consider the problem of finding the square root of a complex number such as When

you use the four basic operations (addition, subtraction, multiplication, and division), 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 a polar

representation for complex numbers, as shown in Figure 8.6. Specifically, if is a

nonzero complex number, then let be the angle from the positive -axis to the radial line

passing through the point and let be the modulus of So,

and

and you have from which the polar form of a complex

number is obtained.

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

any angle.

Because there are infinitely many choices for the argument, the polar form of a complex

number is not unique. Normally, the values of that lie between and are used,

although on occasion it is convenient to use other values. The value of that satisfies the

inequality

Principal argument

is called the principal argument and is denoted by Arg( ). Two nonzero complex numbers

in polar form are equal if and only if they have the same modulus and the same principal

argument.

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 0cos isin,z 0

a bi rcos rsini,

r a2 b2b rsin ,a rcos,

a bi.ra, bx

a bi

1 i22

i.i 1 i

2.

i.

8.3Imaginary

axis

Realaxis

b

r

a 0

Complex Number: a+ bi

Rectangular 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 the

modulus of and is the argument of z.z

rtan ba.a rcos, b rsin, r a2 b2,

z rcos isin

z a biDefinition of the

Polar Form of a

Complex Number

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19/49

Se ct io n 8. 3 Po la r Form and De Moiv re s Th eo re m 499

SOLUT ION (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 8cos3 isin

3

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

Imaginaryaxis

Realaxis

z = i

2

2

1

( )(c) z= 1 cos + isin

Realaxis

1 2

4

3

2

1

z=2 + 3i

Imaginary

axis

(b) z 13[cos(0.98)

+ isin(0.98)]

Imaginary

axis

Realaxis

[ ) ]4

1

2

z= 1 i

(a) z= 2 cos + isin ( )4(

z 1

cos

2 i sin

2.

2,r 1b 1,a 0

z 13 cos0.98 isin0.98. 0.98.

sinb

r

3

13cos

a

r

2

13

r 13.r2 22 32 13,b 3,a 2

z 2 cos4 isin

4. 4

sin b

r

1

2

2

2.cos

a

r

1

22

2

b rsin ,a rcos

r 2.b 1, then r2 12 12 2,a 1

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20/49


SOLUT ION 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 to

you. (See Exercise 65.)

This theorem says that to multiply two complex numbers in polar form, multiply moduli

and add arguments. To divide two complex numbers, divide moduli and subtract arguments.

(See Figure 8.8.)

z1z2

r1r2cos

1

2 isin

1

2.

sin1

2 sin

1cos

2

cos 1sin

2

cos1

2 cos

1cos

2

sin1sin

2

r1r2cos

1cos

2 sin

1sin

2 i cos

1sin

2

sin1cos

2.

z1z2 r1r2cos1 isin 1cos2 isin 2

z2

z1

z2

r2cos

2 isin

2.z

1 r

1cos

1 isin

1

z 8cos3 isin

3 812 i32 4 43i.sin3 32,cos3 12

Given two complex numbers in polar form

and

the product and quotient of the numbers are as follows.

Product

Quotientz1

z2

r1

r2

cos1

2 isin

1

2, z

2 0

z1z2

r1r2cos

1

2 isin

1

2

z2

r2cos

2 isin

2z

1 r

1cos

1 isin

1

THEOREM 8.4

Product and Quotient

of Two Complex Numbers

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21/49


Figure 8.8

Find and for the complex numbers

and

SOLUT ION 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

3

6

1

6

i.z1

52

2

52

2i

z1

z2

513

cos

4

6 isin

4

6 15cos

12 isin

12

z1z2

513cos

4

6 isin

4

6 5

3cos 512 isin

5

12

z2,z

1

z2

1

3cos

6 isin

6.z

1 5

cos

4 isin

4

z1z

2z1z2

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

Imaginary

axis

Realaxis

z1z2

r1r2

r2

r1

z1 z2

11

2

2

1 2To divide z and z :

Divide moduli and subtract arguments.

Imaginary

axis

Realaxis

z1z2

r1r2

r2r1

z1

z2

1

1

2

2+

1 2To multiply z and z :

Multiply moduli and add arguments.

multiply

add add

subtract subtract

divide

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DeMoivres 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

Similarly,

This pattern leads to the next important theorem, named after the French mathematician

Abraham DeMoivre (16671754). You are asked to prove this theorem in Review

Exercise 85.

Find and write the result in standard form.

SOLUT ION First convert to polar form. For

and

which implies that So,

By DeMoivres Theorem,

40961 i 0 4096.

4096cos 8 isin8

212cos 1223 isin122

3

1 3 i12 2cos 23 isin2

312

1 3 i 2cos 23 isin2

3. 23.

tan 3

1 3r 12 32 2

1 3 i,

1 3 i12E X A M P L E 4 Raising a Complex Number to an Integer Power

z5 r5cos 5 isin 5.z

4 r

4

cos 4

isin 4

z3 rcos isin r2cos2 isin 2 r3cos 3 isin3.

z2 rcos i sinrcos isin r2cos 2 isin2

z rcos isin

If and is any positive integer, then

zn rncos n isin n.

nz rcos isinTHEOREM 8.5

DeMoivres Theorem

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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 polynomial such ashas 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 complex

number is defined as follows.

DeMoivres Theorem is useful in determining roots of complex numbers. To see how

this is done, let be an th root of where

and

Then, by DeMoivres Theorem you have and because

it follows that

Now, because the right and left sides of this equation represent equal complex numbers, you

can 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 some

integer which implies that

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

next theorem.

w

2k

n.

k, n 2k,

s nrr2.n

s nr,sn r,

sncos n isin n rcos isin.

wn z,wn sncos n isin n,

z rcos icos .w scos isin

z,nw

n

x1 3 i

2.x

1 3i

2,x 1,

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

px x6 1nn

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

z wn a bin.

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.

nrcos 2kn isin 2k

n nn

z rcos isinn,THEOREM 8.6

The nth Roots of a

Complex Number

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24/49


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

the 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 roots

are equally spaced around the circle, because successive th roots have arguments that

differ byYou have already found the sixth roots of 1 by factoring and the Quadratic Formula. Try

solving the same problem using Theorem 8.6 to see if you get the roots shown in Figure

8.10. When Theorem 8.6 is applied to the real number 1, the th roots have a special

namethe th roots of unity.

Figure 8.10

Determine the fourth roots of

SOLUT ION In polar form, you can write as

so that Then, by applying Theorem 8.6, you have

cos8 k

2 isin

8

k

2.i14 41 cos24

2k

4 isin2

4

2k

4

r 1 and 2.

i 1cos 2 isin

2i

i.

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

Imaginaryaxis

Realaxis

The Sixth Roots of Unity

11

1

1

1

1

2

2

2

2

2

2

2

2

3

3

3

3

+

+

i

i

i

i

n

n

2n.

n

nnrnr,n

n

k 0.

2n

n

n 2

k n,

n 1,k

rn

2

2

n

n

The nth Roots of a

Complex Number

Realaxis

Imaginary

axis

Figure 8.9

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25/49

In Exercises 14, express the complex number in polar form.

1. 2.

3. 4. Imaginaryaxis

Realaxis

3i

1

11

2

3

Imaginaryaxis

Realaxis

6 1

2

2

23456

3

3Imaginary

axis

Realaxis

1 + 3i

1 2

1

2

3

1

Imaginaryaxis

Realaxis

1

1

2

2

2 2i



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 and

is multiplied by 4, the result is of the form

Figure 8.11

Imaginaryaxis

Realaxis

13

8

8

98

813

8

98

cos + isin

58

58

cos + isin

cos + isin

cos + isin

2 2k.1388, 58, 98,

z4

cos13

8 isin

13

8

z3

cos9

8 isin

9

8

z2

cos5

8 isin

5

8

z1

cos

8 isin

8

k 0, 1, 2,

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In Exercises 516, represent the complex number graphically, and

give 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 1726, represent the complex number graphically, and

give the standard form of the number.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

In Exercises 2734, perform the indicated operation and leave the

result in polar form.

27.

28.

29.

30.

31.

32.

33.

34.

In Exercises 3544, use DeMoivres Theorem to find the indicated

powers of the complex number. Express the result in standard form.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

In Exercises 4556, (a) use DeMoivres 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 5764, 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 numbers

andwith prove that

z1

z2

r1

r2cos

1

2 isin

1

2.

z2

0,z2 r2cos 2 isin2,z 1 r1cos 1 isin 1

x4 i 0x3 64i 0

x4 81 0x5 243 0

x3 27 0x3 1 0

x4 16i 0x4 i 0

42 1 i

1252 1 3i

625i

25i

32cos 56 isin5

6

16cos4

3 isin

4

3

9cos 23 isin2

3

16cos 3 isin

3

5cos 32

isin3

2

4

2cos 2

isin

2

8cos 54 isin

5

4103cos 56 isin

5

64

5cos 9 isin

93

1 3i33 i71 i102 2i61 i4

9cos34 isin34

5cos4 isin4

12cos3 isin3

3cos6 isin6

cos53 isin53

cos isin

2cos23 isin23

4[cos56 isin56]

3cos 3 isin

31

3cos 23 isin

2

30.5cos isin0.5cos isin

34cos

2 isin

26cos

4 isin

4

3cos 3 isin

34cos

6 isin

6

6cos isin 7cos 0 isin 0

6cos 56 isin5

64cos3

2 isin

3

2

8cos 6 isin

63.75cos

4 isin

4

3

4cos 74 isin

7

43

2 cos5

3 isin

5

3

5cos 34 isin3

42cos

2 isin

2

5 2i1 2i

22 i3 3 i

2i6i

523 i 21 3i2 2i2 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 the

following.

(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 by

the absolute value of the complex number

is defined as

72. Geometrically, the th roots of any complex number are all

equally spaced around the unit circle centered at the origin.

zn

a bi a2 b2.z a bibi2 b2,

bi

ei 1.

z rei.z rei,

z rei.

z a bi

ei cos isin.

exx i

cos x 1 x2

2!

x4

4!

x6

6! . . . .

sin x xx3

3!

x5

5!

x7

7! . . .

ex 1 xx2

2!

x3

3!

x4

4! . . .

x

x,ex,

i?z

i?z

ziiz,z,

z rcos isin 2cos

6 isin

6.

z rcos isin .

z rcos isin

zz, z 0

zz

zz

z rcos isin.

z rcos isin

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Se ct io n 8 .4 Co mp le x Ve c to r Sp ac es an d In ne r P ro du ct s 509

Complex Vector Spaces and Inner ProductsAll the vector spaces you have studied so far in the text have been real vector spaces

because the scalars have been real numbers. A complex vector space is one in which the

scalars are complex numbers. So, if 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 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 performed

component by component.

Let

and

be vectors in the complex vector space Determine each vector.

(a) (b) (c)

SOLUT ION (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 iu 31 2i, 3 i 5 i2 i, 4

2 iv 2 i1 2i, 3 i 5i, 7 i.

2 i3 i 7 i,2 i1 2i 5i

v u 1 2i3 i 2 i

4 1 3i

7 i.v u

3v 5 iu2 ivv u

C2.

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

E X A M P L E 1 Vector Operations in Cn

CnRn,

v a1 b

1i

a2 b

2i

...an b

ni.

Cn

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

Cn

nCnRnc

1,c2, . . . , c

m

c1v1 c

2v2 c

mvm

v1, v

2, . . . , v

m

8.4

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Many of the properties of are shared by For instance, the scalar multiplicative

identity 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

SOLUT ION 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 Cnby a Basis

v1, v

2, v

3c

1 c

2 c

3 0,

c3i 0.

c2i 0

c1 c2i 0

c1 c2i, c2i,c3i 0, 0, 0

c1i ,0, 0 c2i, c2i, 0 0, 0, c3i 0, 0, 0

c1v1 c

2v2 c

3v3 0, 0, 0

0,

S

v1, v

2, v

3C3

C3.

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.C

n

Cn.Rn

v1 v2 v3

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SOLUT ION 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. Forinstance, the set of complex matrices with matrix addition and scalar multiplication

forms a complex vector space. Example 4 describes a complex vector space in which the

vectors are functions.

Consider the set of complex-valuedfunctions of the form

where and are real-valued functions of a real variable. The set of complex numbers

form 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 S.

af1x bf

2x ibf

1x af

2x

cfx a bif1x if

2x

c a biS

S,

f1x g

1x if

2x g

2x.

fx gx f1x if

2x g

1(x i g

2x

S,

f2f1

fx f1x if

2x

S

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

mnCn,

2, i, 2 i.

v 1 2iv1 v

2 1 2iv

3.

c3

2 i

i 1 2i .c

1

2 i

i 1 2i,

c2 1,

c3i 2 i

c2i i

c1 c

2i 2

2, i, 2 i,

c1 c

2i, c

2i, c

3i

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 the

standard inner (or dot) product in .

Determine the Euclidean inner product of the vectors

and

SOLUT ION

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

PROOF The proof of the first property is shown below, and the proofs of the remaining properties

have been left to you. Let

and v v1, v

2, . . . , v

n.u u

1, u

2, . . . , u

n

Cn

3 i

2 i1 i 02 i 4 5i0

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 C3

Rnvu

Rn

,

Cn

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

u v u1v1 u

2v2 u

nvn.

vuCn.vuDefinition of the Euclidean

Inner 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 0

u kv ku vku v ku vu v w u w v wu v v u

kCnwu, v,THEOREM 8.7

Properties of the

Euclidean Inner Product

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Then

You will now use the Euclidean inner product in 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 as

Determine the norms of the vectors

and

and find the distance between and

SOLUT ION The norms of and are expressed as follows.

2 5 012 7

12 12 22 12 02 0212v v12 v22 v3212

5 0 4112 46

22 12 02 02 42 5212u u12 u22 u3212

vu

v.u

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

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

du,v u1 v12 u2 v22 . . . un vn212.u u12 u22 . . . un212

CnCnCn

u v.

u1v1 u

2v2 u

nvn

v1u1 v

2u2 . . . v

nun

v1u1 v

2u2 . . . v

nun

v u v1u1 v2u2 . . . vnun

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

The Euclidean distance between and is

du, v u v.

vu

u u u12.

uCnuDefinitions of the

Euclidean Norm

and Distance in Cn

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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 inner

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.

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

is a complex inner product.

SOLUT ION 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 u

2 0.u,u 0

u,u u1u1 2u2u2 u12 2u22 0ku,v ku

1v

1 2ku

2v

2 ku

1v1 2u

2v2 ku,v

u,w v,w

u1w

1 2u

2w

2 v

1w

1 2v

2w

2

u v,w u1 v

1w

1 2u

2 v

2w

2

u1v1 2u

2v2 u,vv

1u1 2v

2u2v,u

u, v u1v1 2u2v2

C2

.v v1, v2u u1, u2

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

Cn

1 5 4112 47.

12 02 22 12 42 5212 1, 2 i, 4 5i

du, v u v

vu

Let and be vectors in a complex vector space. A function that associates and with

the complex number is called a complex inner product if it satisfies the following

properties.

1.

2.

3.

4. and if and only if u 0.u,u 0u,u 0ku,v ku,vu v,w

u,w

v,w

u,v v,u

u,vvuvuDefinition of a

Complex Inner Product

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In Exercises 18, perform the indicated operation using

and

1. 2.

3. 4.

5. 6.

7. 8.

In Exercises 912, determine whether is a basis for

9.

10.

11.

12.

In Exercises 1316, express as a linear combination of each of the

following basis vectors.

(a)

(b)

13. 14.

15. 16.

In Exercises 1724, determine the Euclidean norm of

17. 18.

19. 20.

21. 22.

23.

24.

In Exercises 2530, determine the Euclidean distance between

and

25.

26.

27.

28.

29.

30.

In Exercises 3134, determine whether the set of vectors is linearly

independent or linearly dependent.

31.

32.

33.

34.

In Exercises 3538, determine whether the function is a complex

inner product, where and

35.

36.

37.

38.

In Exercises 3942, use the inner product to

find

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 such

that is a basis for

In Exercises 4549, 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 a

complex number. How are and related?

In Exercises 51 and 52, use the inner product

where

and

to find

51.

52. v i3i2i

1u 1

1 i

2i

0

v 101 2i

iu 0

1

i

2i

u, v.

v v11v21

v12

v22

u u11u21

u12

u22

u,v u

11v11 u

12v12 u

21v21 u

22v22

u, kvu,vku,v

u 0.u u 0

u u 0u kv ku v

ku v ku vu v w u w v w

kCnwu, v,

C3.v1, v

2,v

3

v3

v2 1, 0, 1.v

1 i, i, i

z3?z

1,z

2,

C3,v1, v

2, v

3

v3 z1,z2,z3v2 i, i, 0.v1 i, 0, 0

v 2 3i, 2u 4 2i, 3

v 3 i, 3 2iu 2 i, 2 i

v 2 i, 2iu 3 i, i

v i, 4iu 2i, i

u, v.u,v u

1v1 2u

2v2

u,v u1v1 u

2v2

u,v 4u1v1 6u2v2

u,v u1 v

1 2u

2 v

2

u,v u1 u

2v2

v v1, v

2.u u

1, u

2

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, iu 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, 0v 1, 2 i, i

v 2 3i, 2 3iv 36 i, 2 i

v 1, 0v i, i

v.

v i, i, iv i, 2 i, 1

v 1 i, 1 i, 3v 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 iw uu iv 2iw

6 3iv 2 2iwu 2 iv

iv 3w1 2iw

4iw3u

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 5558, the linear transformation is shown

by 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 indicated

composition, where and are the matrices below.

and

61.

62.

63. Determine which of the sets below are subspaces of the vector

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 are real.

(d) The set of diagonal matrices.

64. Determine which of the sets below are subspaces of the vector

space of complex-valued functions (see Example 4).

(a) The set of all functions satisfying

(b) The set of all functions satisfying

(c) The set of all functions satisfying


statement is true or false. If a statement is true, give a reason or cite

an appropriate statement from the text. If a statement is false,



65. Using the Euclidean inner product of and in

66. The Euclidean form of in denoted by is u u2.uCnu

u v u1v1 u2v2 . . . u

nvn.

Cn,vu

fi fi.ff0 1.ffi 0.f

2 2

2 2

A TA.A2 22 2

2 2

T1 T2

T2 T1

T2 ii

i

iT1 0

i

i

0

T2

T1

v i, i

w 1 i

1 i

iv

2

5

0,A

0

i

0

1

i

i

1

1

0,

w 2

2i

3iv 2 i3 2i,A

1

i

i

0

0

i,

w

1

1v

i

01 i

,A

0

i

i

0

1

0,

w 00v 1 i

1 i,A 1

i

0

i,w.vTv Av.

T: CmCn

Ti, 0 2 i, 1, T0, i 0, i

T1, 0 2 i, 1, T0, 1 0, i

T: CmCn

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36/49

Se c ti on 8 .5 Un it ary and He rm it ia n Matr ic es 517

Unitary and Hermitian Matrices

Problems involving diagonalization of complex matrices and the associated eigenvalue

problems require the concepts of unitary and Hermitian matrices. These matrices roughly

correspond to orthogonal and symmetric real matrices. In order to define unitary and

Hermitian matrices, the concept of the conjugate transpose of a complex matrix must first

be introduced.

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 the

transpose of the matrix, as shown in the following example.

Determine * for the matrix

SOLUT ION

Several properties of the conjugate transpose of a matrix are listed in the following

theorem. The proofs of these properties are straightforward and are left for you to supply in

Exercises 5558.

A* AT 3 7i02i

4 i

A 3 7i2i 04 i 3 7i2i 04 i

A 3 7i2i0

4 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 ofA.A

AT

A

AA,Definition of the

Conjugate Transposeof 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.

SOLUT ION 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* A1.

AA* 1

21 i

1 i

1 i

1 i1

2 1 i

1 i

1 i

1 i 1

44

0

0

4 1

0

0

1 I2,

A 1

2 1 i

1 i

1 i

1 1A

E X A M P L E 2 A Unitary Matrix

A1 A

A1 AT.A

A complex matrix is unitaryif

A1 A*.

ADefinition of

Unitary Matrix

An complex matrix is unitary if and only if its row (or column) vectors form an

orthonormal 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

SOLUT ION 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, r3r2 r3 0.r1 r3 0

0.i

2

3

i

2

3

1

2

3

1

2

3

12 i

3 1 i

2 i

3 1

21

3

r1 r2 12 i

3 1 i

2 i

3 1

21

3r2

r1r3r2

14 2

4

1

412 1.

121

2 1 i

2 1 i

2 1

21

212

121

2 1 i

2 1 i

2 1

21

212

r1 r1 r112

r1

r3 5i215,3 i

215,

4 3i

215

r2 i3,i

3,

1

3

r1 12,1 i

2,

1

2r3r1, r2,

A 1

2

i

35i

215

1 i

2

i

33 i

215

1

2

1

34 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 a

matrix is called Hermitian after the French mathematician Charles Hermite (18221901).

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, 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 the entry

in the th row and the th column.

Which matrices are Hermitian?

(a) (b)

(c) (d) 1

2

3

2

0

1

3

1

4

3

2 i

3i

2 i

0

1 i

3i

1 i

0

0

3 2i

3 2i

41

3 i

3 i

i

E X A M P L E 4 Hermitian Matrices

ij

ajijiaij

AA

n n.

A a1b1 b2ib1 b2i

d1.

AA A*.A

a1 a2ib1 b2ic1 c2i

d1 d2i.

a1 a2i

b1 b2i

c1 c2i

d1 d2i

A* AT

A

A a1 a2ic1 c2ib1 b2i

d1 d2i

A.2 2

A square matrix is Hermitian if

A A*.

ADefinition of a

Hermitian Matrix

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SOLUT ION (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 first

column.

(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 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 multiplied by

the 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 matrix

are 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.A

THEOREM 8.10

The Eigenvalues of a

Hermitian Matrix

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Find the eigenvalues of the matrix

SOLUT ION 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 a

real 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

3i2 i

5,1

21i6 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

v1v2

v3

0

0

0

3

2 i

3i

2 i

1 i

3i

1 i

v1v2v3

0

0

0

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

32 2 2 i2 i 3i 3

IA

3

2 i

3i

2 i

1 i

3i

1 i

A

A 3

2 i

3i

2 i

0

1 i

3i

1 i

0A.

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 square

matrix 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 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 (and real)

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 the

Spectral 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 inner

products and For example,v2 v3.v1 v3,v1 v2,

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

v2v1

12,v1*v2 0

2 1v1*v2 02v1*v2 1v1*v2 0

v21v2 1v1v2 v1v2 1v1Av1

v22v2 2v1Av2 v1v2 v1v2 v1Av1

v2.Av1Av2 2v2,Av1 1v12.1

v2v1

APPP

AP1 PP

P1AP

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 thematrix math menu calculates the eigenvalues of the matrix and the eigVc keygives the corresponding eigenvectors.

A,Te chnologyNote

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

C h C l V S

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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 eigenvectors corresponding

to the same eigenvalue may not be orthogonal. Once any set of linearly independent

eigenvectors is obtained for an eigenvalue, however, the Gram-Schmidt orthonormalization

process can be used to find an orthogonal set.

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

SOLUT ION The eigenvectors of are shown after Example 5. Form the matrix by normalizing these

three 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 you

obtain

*

where and are the eigenvalues ofA.21, 6,

AP 1

0

0

06

0

00

2PPAAPP

P

1

71 2i

71

7

1 2i

7286 9i

72813

728

1 3i

402 i

405

40P

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 i

3i

2 i

0

1 i

3i

1 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 11 21i 1 2i6 9i 13

v1 v2 11 21i 1 2i6 9i 113

Se c ti on 8 5 Un it ary and He rm it ia n Matr ic es 525

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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 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 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.

A

AA* 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

P

P

AA

Comparison of Symmetric

and Hermitian Matrices


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

1. 2.

3. 4.

5.

6.

7. 8.

In Exercises 912, use a graphing utility or computer software

program to find the conjugate transpose of the matrix.

9.

10.

11.

12. A 2 i

0

i

1 2i

1

2 i

2 i

4

1

2i

i

0

2i

1 i

1

2i

A 1 i

2 i

1 i

i

0

1

i

2 i

1

0

2

1

i

2i

4i

0

A i

0

1 2i

1

1 i

2i

1 2i

i

i

A

1 i

2i

0

i2 i

1

2 i2i

A

2

5

0

i

3i

6 iA

7 5i

2i

4

A 2 i3 i3 i

2

4 5i

6 2i

A 0

5 i

2 i

5 i

6

4

2i

4

3

A 4 3i2 i2 i

6iA 0

2

1

0

A 1 2i12 i

1A i

2

i

3i


332600_08_5.qxp 4/17/08 11:31 AM Page 526


45/49


In Exercises 1316, explain why the matrix is not unitary.

13. 14.

15.

16.

In Exercises 1722, determine whether is unitary by calculating

*.

17. 18.

19. 20.

21. 22.

In Exercises 2326, (a) verify that is unitary by showing that its

rows are orthonormal, and (b) determine the inverse of

23. 24.

25.

26.

In Exercises 2734, 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 4549, find a unitary matrix that diagonalizes the

matrix

45. 46. A 02 i2 i

4A 0

i

i

0

A.

P

A 1

0

0

4

i

0

1 i

3i

2 i

A 2

i

2

i

2

i

2

2

0

i

2

0

2

A 02 i2 i

4A 3

1 i

1 i

2

A 3i

i

3A 0

i

i

0

A.

A 1

2 i

5

2 i

2

3 i

5

3 i

6

A 12 i2 i

2

3 i

3 i

A 000

0A 1

0

0

1

A 0

2 i

0

i

i

1

1

0

0A

0

2 i

1

2 i

i

0

1

0

1

A i0 0iA 0i i0

A

A

0

1 i6

2

6

1

0

0

0

1 i3

1

3

A 1

223 i3 i 1 3 i

1 3 i

A 1 i

2

1

2

1 i

2

1

2A

4

5

3

5

3

5

4

5

i

iA.

A

A 4

5

3

5i

3

5

4

5iA

i

2i

2

0

i

3i

3i

3

i

6i

6

i

6

A

i

2i

2

i

2

i

2

A i0

0

i

A 1 i1 i1 i

1 iA 1 i

1 i

1 i

1 i

AA

A

A

1

2

i

3

12

1

2

1

3

12

1 i

2

i

3

1

i2

A 1 i

2

0

0

1

i

2

0

A 1

i

i

1A i

0

0

0

Se c ti on 8 5 Un it ary and He rm it ia n Matr ic es 527

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


46/49


47.

48.

49.

50. Let be a complex number with modulus 1. Show that the

matrix is unitary.

In Exercises 5154, use the result of Exercise 50 to determine

and such that is unitary.

51. 52.

53. 54.

In Exercises 5558, prove the formula, where and are

complex matrices.

55. 56.

57. 58.

59. Let be a matrix such that Prove that is

Hermitian.

60. Show that det where is a matrix.

In Exercises 61 and 62, assume that the result of Exercise 60 is true

for 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 matrix

and is real and skew-symmetric.

(c) Prove that every complex matrix can be written as

where 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 the

vector 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 Hermitian

nor unitary.

(f ) Find the eigenvalues and corresponding eigenvectors of

your 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

A1 A*A

AA*.A In

i01

i

2 2

2 22 2

n n

n n

n n

n n

CBA B iC,

A i2 i2

1 2i

CBA B iC,

An n

C

BA B iC,

A 21 i 1 i

3

C

BA B iC,A

detA 1.A

detA* detA.

2 2AA detA,

iAA* A O.AAB*

B*A*kA*

kA*

A B* A* B*A** A

n nBA

A 1

2 ab

6 3i

45

cA 12 ib ac

A 1

2 3 4i

5b

a

cA

1

2 1

b

a

c

Ac

a, b,

A 1

2z

iz

z

izA

z

A

1

0

0

0

1

1 i

0

1 i

0

A 42 2i2 2i

6

A

2

i

2

i

2

i

2

2

0

i

2

0

2


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p p p

Review ExercisesCHAPTER 8

In Exercises 16, perform the operation.

1. Find

2. Find

3. Find

4. Find

5. Find

6. Find

In Exercises 714, find all zeros of the polynomial function.

7. 8.

9. 10.

11. 12.

13.

14.

In Exercises 1522, perform the operation using

and

15. 16. 17.

18. 19. 20.

21. 22.

In Exercises 2328, perform the operation using

and

23. 24. 25.

26. 27. 28.

In Exercises 2932, perform the indicated operation.

29. 30.

31. 32.

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

33.

34.

In Exercises 3540, determine the polar form of the complex

number.

35. 36. 37.

38. 39. 40.

In Exercises 4146, find the standard form of the complex number.

41.

42.

43. 44.

45. 46.

In Exercises 4750, perform the indicated operation. Leave the

result in polar form.

47.

48.

49.

50.

In Exercises 5154, find the indicated power of the number andexpress the result in polar form.

51. 52.

53. 54.

In Exercises 5558, express the roots in standard form.

55. Square roots:

56. Cube roots:

57. Cube roots:i

27cos

6 isin

6

25cos 23 isin2

3

5cos 3 isin

34

2cos 6 isin

67

2i31 i4

4cos 4 isin 4

7 cos 3

lineal capitulo 8

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