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So far we have coded the complex number
z=x+jyz=x+jy
with the Cartesian
pair
(x,y)(x,y) and with the polar pair (r∠θ)(r∠θ).
We now show how the complex
number
z
z may be coded with a two-dimensional vector
z
z and show how this
new code may be used to gain insight about complex numbers.
Coding a Complex Number as a Vector. We code the complex
number z=x+jyz=x+jy with the two-dimensional vector z=xyz=xy:
x
+
j
y
=
z
⇔
z
=
x
y
.
x
+
j
y
=
z
⇔
z
=
x
y
.
(1)We plot this vector as in Figure 1. We say that the vector zz belongs to a “vector space.” This means that vectors may be added and scaled according to the rules
z
1
+
z
2
=
x1+x2y1+y2
z
1
+
z
2
=
x1+x2y1+y2
(2)
a
z
=
a
x
a
y
.
a
z
=
a
x
a
y
.
(3)Furthermore, it means that an additive inverse -z-z, an additive identity 00,
and a multiplicative identity 1 all exist:
z
+
(
-
z
)
=
0
z
+
(
-
z
)
=
0
(4)The vector
0
0 is 0=000=00.
Prove that vector addition and scalar multiplication satisfy
these properties of commutation, association, and distribution:
z
1
+
z
2
=
z
2
+
z
1
z
1
+
z
2
=
z
2
+
z
1
(6)
(
z
1
+
z
2
)
+
z
3
=
z
1
+
(
z
2
+
z
3
)
(
z
1
+
z
2
)
+
z
3
=
z
1
+
(
z
2
+
z
3
)
(7)
a
(
b
z
)
=
(
a
b
)
z
a
(
b
z
)
=
(
a
b
)
z
(8)
a
(
z
1
+
z
2
)
=
a
z
1
+
a
z
2
.
a
(
z
1
+
z
2
)
=
a
z
1
+
a
z
2
.
(9)Inner Product and Norm. The inner product between two vectors
z
1
z
1
and
z
2
z
2
is defined to be the real number
(z1,z2)=x1x2+y1y2.(z1,z2)=x1x2+y1y2.
(10)We sometimes write this inner product as the vector product (more on this
in Linear Algebra)
(
z
1
,
z
2
)
=
z
1
T
z
2
=
[
x
1
y
1
]
x2y2
=
(
x
1
x
2
+
y
1
y
2
)
.
(
z
1
,
z
2
)
=
z
1
T
z
2
=
[
x
1
y
1
]
x2y2
=
(
x
1
x
2
+
y
1
y
2
)
.
(11)
Prove (z1,z2)=(z2,z1).(z1,z2)=(z2,z1).
When z1=z2=zz1=z2=z, then the inner product between zz and itself is the
norm squared of zz:
|
|
z
|
|
2
=
(
z
,
z
)
=
x
2
+
y
2
.
|
|
z
|
|
2
=
(
z
,
z
)
=
x
2
+
y
2
.
(12)These properties of vectors seem abstract. However, as we now show, they
may be used to develop a vector calculus for doing complex arithmetic.
A Vector Calculus for Complex Arithmetic. The addition of two complex numbers
z
1
z
1
and
z
2
z
2
corresponds to the addition of the vectors
z
1
z
1
and
z
2
z
2
:
z
1
+
z
2
⇔
z
1
+
z
2
=
x1+x2y1+y2
z
1
+
z
2
⇔
z
1
+
z
2
=
x1+x2y1+y2
(13)The scalar multiplication of the complex number
z
2
z
2
by the real number
x
1
x
1
corresponds to scalar multiplication of the vector
z
2
z
2
by
x
1
x
1
:
x
1
z
2
⇔
x
1
x
2
y
2
=
x
1
x
2
x
1
y
2
.
x
1
z
2
⇔
x
1
x
2
y
2
=
x
1
x
2
x
1
y
2
.
(14)Similarly, the multiplication of the complex number
z
2
z
2
by the real number
y
1
y
1
is
y
1
z
2
↔
y
1
x
2
y
2
=
y
1
x
2
y
1
y
2
.
y
1
z
2
↔
y
1
x
2
y
2
=
y
1
x
2
y
1
y
2
.
(15)The complex product z1z2=(x1+jy1)z2z1z2=(x1+jy1)z2 is therefore represented as
z
1
z
2
↔
x
1
x
2
-
y
1
y
2
x
1
y
2
+
y
1
x
2
.
z
1
z
2
↔
x
1
x
2
-
y
1
y
2
x
1
y
2
+
y
1
x
2
.
(16)This representation may be written as the inner product
z
1
z
2
=
z
2
z
1
↔
(
v
,
z
1
)
(
w
,
z
1
)
z
1
z
2
=
z
2
z
1
↔
(
v
,
z
1
)
(
w
,
z
1
)
(17)where
v
v and
w
w are the vectors v=x2-y2v=x2-y2 and w=y2x2w=y2x2. By defining the
matrix
x
2
-
y
2
y
2
x
2
,
x
2
-
y
2
y
2
x
2
,
(18)we can represent the complex product z1z2z1z2 as a matrix-vector multiply (more
on this in Linear Algebra):
z
1
z
2
=
z
2
z
1
↔
x
2
-
y
2
y
2
x
2
x
1
y
1
.
z
1
z
2
=
z
2
z
1
↔
x
2
-
y
2
y
2
x
2
x
1
y
1
.
(19)With this representation, we can represent rotation as
z
e
j
θ
=
e
j
θ
z
↔
cos
θ
-
sin
θ
sin
θ
cos
θ
x
1
x
2
.
z
e
j
θ
=
e
j
θ
z
↔
cos
θ
-
sin
θ
sin
θ
cos
θ
x
1
x
2
.
(20)We call the matrix cosθ-sinθsinθcosθcosθ-sinθsinθcosθ a “rotation matrix.”
Call R(θ)R(θ) the rotation matrix:
R
(
θ
)
=
cos
θ
-
sin
θ
sin
θ
cos
θ
.
R
(
θ
)
=
cos
θ
-
sin
θ
sin
θ
cos
θ
.
(21)Show that R(-θ)R(-θ) rotates by (-θ)(-θ). What can you say about R(-θ)wR(-θ)w when
w=R(θ)zw=R(θ)z?
Represent the complex conjugate of
zz as
z
*
↔
a
b
c
d
x
y
z
*
↔
a
b
c
d
x
y
(22)and find the elements a,b,ca,b,c, and
d
d of the matrix.
Inner Product and Polar Representation. From the norm of a vector, we derive a formula for the magnitude of
z
z in the polar representation
z=rejθz=rejθ :
r
=
(
x
2
+
y
2
)
1
/
2
=
|
|
z
|
|
=
(
z
,
z
)
1
/
2
.
r
=
(
x
2
+
y
2
)
1
/
2
=
|
|
z
|
|
=
(
z
,
z
)
1
/
2
.
(23)If we define the coordinate vectors e1=10e1=10 and e2=01e2=01, then we
can represent the vector
z
z as
z=(z,e1)e1+(z,e2)e2.z=(z,e1)e1+(z,e2)e2.
(24)See Figure 2. From the figure it is clear that the cosine and sine of the
angle
θθ are
cos
θ
=
(
z
,
e
1
)
|
|
z
|
|
;
sin
θ
=
(
z
,
e
2
)
|
|
z
|
|
cos
θ
=
(
z
,
e
1
)
|
|
z
|
|
;
sin
θ
=
(
z
,
e
2
)
|
|
z
|
|
(25)This gives us another representation for any vector zz:
z
=
|
|
z
|
|
cos
θ
e
1
+
|
|
z
|
|
sin
θ
e
2
.
z
=
|
|
z
|
|
cos
θ
e
1
+
|
|
z
|
|
sin
θ
e
2
.
(26)The inner product between two vectors
z
1
z
1
and
z
2
z
2
is now
(
z
1
,
z
2
)
=
[
(
z
1
,
e
1
)
e
1
T
(
z
1
,
e
2
)
e
2
T
]
(
z
2
,
e
1
)
e
1
(
z
2
,
e
2
)
e
2
=
(
z
1
,
e
1
)
(
z
2
,
e
1
)
+
(
z
1
,
e
2
)
(
z
2
,
e
2
)
=
|
|
z
1
|
|
cos
θ
1
|
|
z
2
|
|
cos
θ
2
+
|
|
z
1
|
|
s
i
n
θ
1
|
|
z
2
|
|
sin
θ
2
.
(
z
1
,
z
2
)
=
[
(
z
1
,
e
1
)
e
1
T
(
z
1
,
e
2
)
e
2
T
]
(
z
2
,
e
1
)
e
1
(
z
2
,
e
2
)
e
2
=
(
z
1
,
e
1
)
(
z
2
,
e
1
)
+
(
z
1
,
e
2
)
(
z
2
,
e
2
)
=
|
|
z
1
|
|
cos
θ
1
|
|
z
2
|
|
cos
θ
2
+
|
|
z
1
|
|
s
i
n
θ
1
|
|
z
2
|
|
sin
θ
2
.
(27)It follows that cos(θ2-θ1)=cosθ2cos(θ2-θ1)=cosθ2 cos θ1+sinθ1sin
θ
2
θ1+sinθ1sinθ
2
may be written as
cos(θ2-θ1)=(z1,z2)||z1||
||z2||cos(θ2-θ1)=(z1,z2)||z1||
||z2||
(28)This formula shows that the cosine of the angle between two vectors
z
1
z
1
and
z
2
z
2
, which is, of course, the cosine of the angle of z2z1*z2z1*, is the ratio of the inner
product to the norms.
Prove the Schwarz and triangle inequalities and interpret them:
(
Schwarz
)
(
z
1
,
z
2
)
2
≤
|
|
z
1
|
|
2
|
|
z
2
|
|
2
(
Schwarz
)
(
z
1
,
z
2
)
2
≤
|
|
z
1
|
|
2
|
|
z
2
|
|
2
(29)
(
triangle
)
I
z
1
-
z
2
|
|
≤
|
|
z
1
-
z
3
|
|
+
|
|
z
2
-
z
3
|
|
.
(
triangle
)
I
z
1
-
z
2
|
|
≤
|
|
z
1
-
z
3
|
|
+
|
|
z
2
-
z
3
|
|
.
(30)
(What does "Endorsed by" mean?)
"Reviewer's Comments: 'I recommend this book as a "required primary textbook." This text attempts to lower the barriers for students that take courses such as Principles of Electrical Engineering, […]"