Imaginary Number

$⟨$ Definition A-12.2 $⟩$ Imaginary Number

$i$ is defined to be an imaginary number that has the property: $i^{2} = - 1$
Complex Number is a number of the form $a + b i$ , where $a$ and $b$ are real numbers (e.g., $a = 5.6 + 4.3 i)$
Real Number is a number that does not involve any imaginary number (e.g., $a = 13.4$ )
$\bar{a}$ is a Conjugate of $a$ if $a$ and $\bar{a}$ have the same real number part and an opposite-signed imaginary number part
(e.g., $a = 3 + i \cdot 3.4$ , $\bar{a} = 3 - i \cdot 3.4$ )
Hermitian Vector is a vector where the 2nd half of its elements is the complex conjugate of the 1st half in reverse order, as illustrated by the $n$ -dimensional vector below:
$\vec{v} = (v_{1}, v_{2}, v_{3}, \dots, v_{\frac{n}{2} - 1}, v_{\frac{n}{2}}, {\bar{v}}_{\frac{n}{2}}, {\bar{v}}_{\frac{n}{2} - 1}, \dots, {\bar{v}}_{3}, {\bar{v}}_{2}, {\bar{v}}_{1})$