Orthonormality

Orthonormality is a special case of orthogonality.

Mathematical definition

Let vector space $V$ be an inner product space.

If for $u,v\in V$

$$ \begin{cases} \langle u, v \rangle=0 & \text{(orthogonality)} \\ \langle u, u \rangle=1 \land \langle v, v \rangle=1 & \text{(normality)} \end{cases} $$

Then $u,v$ are orthonormal to each other.

Orthonormal vector list

A list of nonzero vectors $(e_1,\ldots,e_m) \in V$ is orthonormal if

$$ \langle e_i, e_j \rangle = \delta_{i,j} \Leftarrow i,j \in \{1,m\} $$

where $\delta_{ij}$ is the Kronecker delta function, i.e., $\delta_{ij} = \text{ if } i=j \text{ otherwise } 0$.

which means:

$$\begin{gathered} \langle e_i, e_j \rangle = 0 \Leftarrow 1\le i\neq j \le m & \text{(orthogonality)}\\ \langle e_i, e_i \rangle = 1 \Leftarrow 1\le i \le m & \text{(normality)} \end{gathered}$$

Orthonormal basis

Any orthonormal vector list of length $dim(V)$ is an orthonormal basis for $V$.

Which means:

If $(e_1,\ldots,e_n)$ is an orthonormal vector list for $V \in \mathbb{F}^n$

Then

$$ \forall v = a_1 e_1 + \cdots + a_n e_n $$

Property

$$ \forall v\in V \begin{cases} v=\sum_{k=1}^n \langle v,e_k\rangle e_k \\ \lVert v\rVert^2=\sum_{k=1}^n |\langle v,e_k\rangle|^2 \end{cases} $$

Proof:

$$\begin{gathered} \because v = a_1 e_1 + \cdots + a_n e_n & \text{(1)} \\ \therefore \forall k \in \{1, \cdots, n\}, \langle v, e_k \rangle= \langle a_1 e_1 + \cdots + a_n e_n , e_k \rangle = a_k \\ \therefore \text{(1) becomes } v = \langle v, e_1 \rangle e_1 + \cdots + \langle v, e_n \rangle e_n \\ \text{i.e., } v=\sum_{k=1}^n \langle v,e_k\rangle e_k \\ \end{gathered}$$

and

$$ \lVert v\rVert^2=\sum_{k=1}^n |a_k|^2=\sum_{k=1}^n |\langle v,e_k\rangle|^2 $$