Equivalence of Definitions of Complex Dot Product
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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 |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Dot and Cross Product