Orthonormality - Revision history

Rongzhou: /* Orthonormal basis */

2026-10-01T13:12:03Z

Orthonormal basis

← Older revision		Revision as of 13:12, 1 October 2026
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	== Orthonormal basis ==		== Orthonormal basis ==

	Any [[#orthonormal vector list\|orthonormal vector list]] of length $dim(V)$ is an '''orthonormal basis''' for $V$.		Any [[#orthonormal vector list\|orthonormal vector list]] of length $dim(V)$ is an '''orthonormal basis''' for $V$.<ref>{{#cite:Q1930}}</ref>

	Which means:		Which means:

Rongzhou: /* Mathematical definition */

2026-10-01T13:11:11Z

Mathematical definition

← Older revision		Revision as of 13:11, 1 October 2026
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	Then $u,v$ are '''orthonormal''' to each other.		Then $u,v$ are '''orthonormal''' to each other.
			<ref>{{#cite:Q1930}}</ref>

	== Orthonormal vector list ==		== Orthonormal vector list ==

Rongzhou at 12:55, 1 October 2026

2026-10-01T12:55:23Z

← Older revision	Revision as of 12:55, 1 October 2026
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		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:

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

		== Orthonormal basis ==

		Any [[#orthonormal vector list\|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:'''

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

		and

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

Rongzhou: Created the wiki page for orthonormality

2026-10-01T10:44:20Z

Created the wiki page for orthonormality

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