3.4a Inner Product

Inner Product

The inner product is a function that takes two vectors and returns a real scalar quantifying how much the vectors align in direction.

For a vector space on $\mathbb{R}$, the inner product of vectors $\vec{x}\in {\mathbb{R}}^n$  and  $\vec{y}\in {\mathbb{R}}^n$ is also called the dot product:
$⟨\vec{x},\vec{y}⟩=\vec{x}\cdot \vec{y}={\vec{x}}^T\vec{y}={\sum }_{i=1}^n{x}_i{y}_i.$
Furthermore, the dot product of two vectors in ${\mathbb{R}}^n$ can be given by the geometric definition,
${\vec{x}}^T\vec{y}=\Vert \vec{x}\Vert \Vert \vec{y}\Vert \cos \theta ,$
where $\theta$ is the angle between the two vectors.

However, for vector spaces on $\mathbb{C}$, the inner product introduces the complex conjugate, which keeps the output scalar on the real line. Therefore, for two complex vectors  $\vec{x}\in {\mathbb{C}}^n$  and  $\vec{x}\in {\mathbb{C}}^n$, the inner product is defined as
$⟨\vec{x},\vec{y}⟩={\vec{x}}^T\vec{y}\text{textasteriskcentered}={\sum }_{i=1}^n{x}_i{y}_i\text{textasteriskcentered}.$

Properties

(Cauchy-Schwartz Inequality) $|{\vec{x}}^T\vec{y}|\le \Vert \vec{x}\Vert \Vert \vec{y}\Vert$.

This follows from the fact that $\cos \theta \le 1$.