Vector Arithmetic

This section explains the basic arithmetic of vector and matrix computations, as well as advanced concepts such as vector/plane projection and the basis of planes (or spaces).

$⟨$ Definition A-10.1 $⟩$ Vector Arithmetic

Addition: Given two $n \times 1$ vectors (i.e., $n$ -dimensional vectors) composed of $n$ numbers each:
$\vec{a} = (a_{0}, a_{1}, \dots, a_{n - 1})$ , $\vec{b} = (b_{0}, b_{1}, \dots, b_{n - 1})$

Vector addition is defined as:
$\vec{a} + \vec{b} = (a_{0} + b_{0}, a_{1} + b_{1}, \dots, a_{n - 1} + b_{n - 1})$
Dot Product: Given two $n$ -dimensional vectors:
$\vec{a} = (a_{0}, a_{1}, …, a_{n - 1})$ , $\vec{b} = (b_{0}, b_{1}, …, b_{n - 1})$

Dot product is defined as:
$⟨ \vec{a}, \vec{b} ⟩ = \vec{a} \cdot \vec{b} = a_{0} b_{0} + a_{1} b_{1} + \dots + a_{n - 1} b_{n - 1}$ .
$\vec{a} \cdot \vec{b}$ can also be expressed as $∥ \vec{a} ∥ ∥ \vec{b} ∥ \cos 𝜃$ , where $𝜃$ is the angle between $\vec{a}$ and $\vec{b}$ . If $\vec{a}$ and $\vec{b}$ point in the same direction, $\vec{a} \cdot \vec{b}$ attains its maximum value ( $∥ \vec{a} ∥ ∥ \vec{b} ∥$ ). If $\vec{a}$ and $\vec{b}$ are orthogonal, then $\vec{a} \cdot \vec{b} = 0$ .
Hadamard Product: Given two $n$ -dimensional vectors:
$\vec{a} = (a_{0}, a_{1}, …, a_{n - 1})$ , $\vec{b} = (b_{0}, b_{1}, …, b_{n - 1})$

Hadamard product is defined as:
$\vec{a} ⊙ \vec{b} = (a_{0} b_{0}, a_{1} b_{1}, \dots, a_{n - 1} b_{n - 1})$
Hermitian Product: Given two $n$ -dimensional complex vectors:
$\vec{a} = (a_{0} + i \cdot a_{0}^{'}, a_{1} + i \cdot a_{1}^{'}, …, a_{n - 1} + i \cdot a_{n - 1}^{'})$
$\vec{b} = (b_{0} + i \cdot b_{0}^{'}, b_{1} + i \cdot b_{1}^{'}, …, b_{n - 1} + i \cdot b_{n - 1}^{'})$

Hermitian product is a dot product with the 2nd operand as a conjugate (§A-11.2):
$⟨ ⟨ \vec{a}, \vec{b} ⟩ ⟩ = \vec{a} \cdot \bar{\vec{b}}$
$= (a_{0} + i \cdot a_{0}^{'}, a_{1} + i \cdot a_{1}^{'}, \dots, a_{n - 1} + i \cdot a_{n - 1}^{'}) \cdot (b_{0} - i \cdot b_{0}^{'}, b_{1} - i \cdot b_{1}^{'}, \dots, b_{n - 1} - i \cdot b_{n - 1}^{'})$

A-10.1 Vector Arithmetic