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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
.
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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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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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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
Distributive property
Use
Commutative property
Distributive 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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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
496 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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)
True or False? In Exercises 42 and 43, 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.
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
Se c ti on 8 .2 Co nju gat es and Di v is io n of Co mp le x Numb ers 497
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498 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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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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500 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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Se ct io n 8. 3 Po la r Form and De Moiv re s Th eo re m 501
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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502 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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Se ct io n 8. 3 Po la r Form and De Moiv re s Th eo re m 503
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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504 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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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
ExercisesSECTION 8.3
Se ct io n 8. 3 Po la r Form and De Moiv re s Th eo re m 505
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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506 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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Se ct io n 8. 3 Po la r Form and De Moiv re s Th eo re m 507
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
True or False? In Exercises 71 and 72, 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.
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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510 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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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 511
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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512 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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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 513
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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514 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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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 515
ExercisesSECTION 8.4
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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516 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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
True or False? In Exercises 65 and 66, 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.
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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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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518 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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Se c ti on 8 .5 Un it ary and He rm it ia n Matr ic es 519
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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520 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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Se c ti on 8 .5 Un it ary and He rm it ia n Matr ic es 521
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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522 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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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524 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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
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Se c ti on 8 .5 Un it ary and He rm it ia n Matr ic es 525
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
ExercisesSECTION 8.5
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
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526 C h ap t er 8 C o mp l ex Ve ct o r Sp a ce s
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
42. The matrix in Exercise 38
43. The matrix in Exercise 39
44. The matrix in Exercise 36
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
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Se c ti on 8 .5 Un it ary and He rm it ia n Matr ic es 527
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
True or False? In Exercises 67 and 68, 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.
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