15.3 Inner products

Definition: inner product

You may have run across inner products , also called dot products , on $\mathbb{R}^n$ before in some of your math or science courses. If not, we define the inner product as follows, given we have some $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$

standard inner product

The standard inner product is defined mathematically as:

$x\cdot y=y^Tx=\begin{pmatrix}{y}_0 & y_1 & \dots & y_{n-1}\\ \end{pmatrix}\left(\begin{array}{c}{x}_0\\ x_1\\ ⋮\\ x_{n-1}\end{array}\right)=\sum_{i=0}^{n-1} x_i{y}_i$

Inner product in 2-d

If we have $x\in \mathbb{R}^2$ and $y\in \mathbb{R}^2$ , then we can write the inner product as

Geometrically, the inner product tells us about the strength of $x$ in the direction of $y$ .

The following characteristics are revealed by the inner product:

• $x\cdot y$ measures the length of the projection of $y$ onto $x$ .
• $x\cdot y$ is maximum (for given $(x)$ , $(y)$ ) when $x$ and $y$ are in the same direction ( $(\theta =0)\implies (\cos \theta =1)$ ).
• $x\cdot y$ is zero when $(\cos \theta =0)\implies (\theta =90°)$ , i.e. $x$ and $y$ are orthogonal .

Inner product rules

In general, an inner product on a complex vector space is just a function (taking two vectors and returning a complexnumber) that satisfies certain rules:

• Conjugate Symmetry: $$x\cdot y=\overline{x\cdot y}$$
• Scaling: $$\alpha x\cdot y=\alpha (x\cdot y)$$
• Additivity: $$x+y\cdot z=x\cdot z+y\cdot z$$
• "Positivity": $$\forall x, x\neq 0\colon x\cdot x> 0$$

orthogonal

We say that $x$ and $y$ are orthogonal if: $$x\cdot y=0$$