Equivalence of Definitions of Complex Dot Product

Theorem

The following definitions of the concept of Complex Dot Product are equivalent:

Definition 1

The dot product of $z_1$ and $z_2$ is defined as:

$z_1 \circ z_2 = x_1 x_2 + y_1 y_2$

Definition 2

The dot product of $z_1$ and $z_2$ is defined as:

$z_1 \circ z_2 = \cmod {z_1} \, \cmod{z_2} \cos \theta$

where:

$\cmod {z_1}$ denotes the complex modulus of $z_1$
$\theta$ denotes the angle between $z_1$ and $z_2$.

Definition 3

The dot product of $z_1$ and $z_2$ is defined as:

$z_1 \circ z_2 := \map \Re {\overline {z_1} z_2}$

where:

$\map \Re z$ denotes the real part of a complex number $z$
$\overline {z_1}$ denotes the complex conjugate of $z_1$
$\overline {z_1} z_2$ denotes complex multiplication.

Definition 4

The dot product of $z_1$ and $z_2$ is defined as:

$z_1 \circ z_2 := \dfrac {\overline {z_1} z_2 + z_1 \overline {z_2} } 2$

where:

$\overline {z_1}$ denotes the complex conjugate of $z_1$
$\overline {z_1} z_2$ denotes complex multiplication.

Proof

Definition 1 equivalent to Definition 3

 $\ds$  $\ds \map \Re {\overline {z_1} z_2}$ Definition 3 of Dot Product $\ds$ $=$ $\ds \map \Re {\paren {x_1 - i y_1} {x_2 + i y_2} }$ Definition of Complex Conjugate $\ds$ $=$ $\ds \map \Re {\paren {x_1 x_2 + y_1 y_2} + i \paren {x_1 y_2 - x_2 y_1} }$ Definition of Complex Multiplication $\ds$ $=$ $\ds x_1 x_2 + y_1 y_2$ Definition of Real Part

$\Box$

Definition 2 equivalent to Definition 3

 $\ds$  $\ds \map \Re {\overline {z_1} z_2}$ Definition 3 of Dot Product $\ds$ $=$ $\ds r_1 r_2 \map \cos {\theta_2 - \theta_1}$ Complex Dot Product in Exponential Form $\ds$ $=$ $\ds \cmod {z_1} \, \cmod {z_2} \map \cos {\theta_2 - \theta_1}$ Definition of Polar Form of Complex Number $\ds$ $=$ $\ds \cmod {z_1} \, \cmod {z_2} \cos \theta$ where $\theta = \theta_2 - \theta_1$ is the angle between $z_1$ and $z_2$

$\Box$

Definition 1 equivalent to Definition 4

 $\ds$  $\ds \frac {\overline {z_1} z_2 + z_1 \overline {z_2} } 2$ Definition 4 of Dot Product $\ds$ $=$ $\ds \frac {\paren {x_1 - i y_1} \paren {x_2 + i y_2} + \paren {x_1 + i y_1} \paren {x_2 - i y_2} } 2$ Definition of Complex Conjugate $\ds$ $=$ $\ds \frac {\paren {\paren {x_1 x_2 + y_1 y_2} + i \paren {x_1 y_2 - x_2 y_1} } + \paren {\paren {x_1 x_2 + y_1 y_2} + i \paren {-x_1 y_2 + x_2 y_1} } } 2$ Definition of Complex Multiplication $\ds$ $=$ $\ds x_1 x_2 + y_1 y_2$ after algebra

$\blacksquare$