Orthogonal complement

Given an inner product space $(𝑉, 𝕂, ⟨ \cdot | \cdot ⟩)$ , the orthogonal complement $𝐴 ⟂$ of a subset $𝐴 \subseteq 𝑉$ is the vector subspace of vectors orthogonal to those $𝐴$ #m/def/linalg

𝐴 ⟂ = {𝑣 \in 𝑉 : (\forall 𝑎 \in 𝐴) [⟨ 𝑎 | 𝑣 ⟩ = 0]}

Proof of subspace

Clearly\Span $⃗ 𝟎 \in 𝐴 ⟂$ . If $𝑣 1, 𝑣 2 \in 𝐴 ⟂$ and $𝛼, 𝛽 \in 𝕂$ then $⟨ 𝑎 | 𝛼 𝑣 1 + 𝛽 𝑣 2 ⟩ = 𝛼 ⟨ 𝑎 | 𝑣 1 ⟩ + 𝛽 ⟨ 𝑎 | 𝑣 2 ⟩ = 0$ for all $𝑎 \in 𝐴$ , and thus $𝛼 𝑣 1 + 𝛽 𝑣 2 \in 𝐴 ⟂$ . Therefore $𝐴 ⟂$ is a subspace.

A slightly more general concept is Dual annihilator.

Properties

Let $𝐴 \subseteq 𝑉$ be an arbitrary subset. Then

$𝐴 ⟂$ is topologically closed
$𝐴 \cap 𝐴 ⟂ = {0}$
$𝐵 \subseteq 𝐴 ⟹ 𝐴 ⟂ \subseteq 𝐵 ⟂$
$𝐴 \subseteq (𝐴 ⟂) ⟂$
If $B 𝜖 (𝑣) \subseteq 𝐴$ for some $𝑣 \in 𝑉$ , then $𝐴 ⟂ = {0}$
$𝐴 ⟂ = (s p a n 𝐴) ⟂$

Proof of 1–6

Note that the orthogonal complement of a singleton ${𝑣}$ can be expressed as a preïmage

{𝑣} ⟂ = (⟨ 𝑣 |) - 1 {0}

Since ${0}$ is closed in $𝕂$ , and the inner product is continuous, it follows ${𝑣} ⟂$ is closed. Now for an arbitrary set $𝐴$ ,

𝐴 ⟂ = ⋂ 𝑎 \in 𝐴 {𝑎} ⟂

which is an intersection of closed sets and is therefore closed, proving ^S1.

Note if $𝑣 \in 𝐴 \cap 𝐴 ⟂$ then $⟨ 𝑣 | 𝑣 ⟩ = 0$ , implying $𝑣 = 0$ by ^IP3, proving ^S2.

Let $𝐵 \subseteq 𝐴 \subseteq 𝑉$ and $𝑣 \in 𝐴 ⟂$ . Then $⟨ 𝑣 | 𝑏 ⟩ = 0$ for any $𝑏 \in 𝐵 \subseteq 𝐴$ , so $𝑣 \in 𝐵 ⟂$ , proving ^S3.

Let $𝑎 \in 𝐴$ . Then by definition $⟨ 𝑎 | 𝑣 ⟩ = 0$ for any $𝑣 \in 𝐴 ⟂$ , so $𝑎 \in (𝐴 ⟂) ⟂$ , proving ^S4.

Without loss of generality $𝑣 = 0$ , for $⟨ 𝐵 𝜖 (𝑣) | 𝑎 ⟩ = 0$ iff $⟨ B 𝜖 (𝑣) - 𝑣 | 𝑎 ⟩ = ⟨ B 𝜖 (0) | 𝑎 ⟩ = 0$ . Now

B 𝜖 (0) = {𝑣 \in 𝑉 : ⟨ 𝑣 | 𝑣 ⟩ < 𝜖}

and

⟨ B 𝜖 (0) | 𝑦 ⟩ = 0 ⟺ ⟨ B 𝜖 (0) | 𝜖 𝑦 2 ‖ 𝑦 ‖ ⟩ = 0

but since $𝜖 𝑦 2 ‖ 𝑦 ‖ \in B 𝜖 (0)$ , it follows from ^IP3 that $𝑦 = 0$ , proving ^S5.

Let $𝑥 \in 𝐴 ⟂$ and $𝑦 \in s p a n 𝐴$ , so $𝑦 = \sum 𝑛 𝑖 = 1 𝜆 𝑖 𝑎 𝑖$ for some ${𝑎 𝑖} 𝑛 𝑖 = 1 \subseteq 𝐴$ . Then

⟨ 𝑥 | 𝑦 ⟩ = ⟨ 𝑥 | 𝑛 \sum 𝑖 = 1 𝜆 𝑖 | 𝑎 𝑖 ⟩ = 0

proving ^S6.

Let $𝑊 \leq 𝑉$ be a vector subspace. Then

$𝑉 = 𝑊 \oplus 𝑊 ⟂$ (Internal direct sum).
$𝑊 = (𝑊 ⟂) ⟂$ .

Proof of 1–2

^V1 follows directly from ^S2, and ^V2 follows directly from ^S4.

Other properties include

The orthogonal complement of an invariant subspace under a unitary operator is invariant

Orthogonal complement

Properties

See also