View source for Orthogonal complement

{{Short description|Concept in linear algebra}}
In the [[mathematics|mathematical]] fields of [[linear algebra]] and [[functional analysis]], the '''orthogonal complement''' of a [[linear subspace|subspace]] <math>W</math> of a [[vector space]] <math>V</math> equipped with a [[bilinear form]] <math>B</math> is the set <math>W^\perp</math> of all vectors in <math>V</math> that are [[orthogonal]] to every vector in <math>W</math>.  Informally, it is called the '''perp''', short for '''perpendicular complement'''. It is a subspace of <math>V</math>.

==Example==

Let <math>V = (\R^5, \langle \cdot, \cdot \rangle)</math> be the vector space equipped with the usual [[dot product]] <math>\langle \cdot, \cdot \rangle</math> (thus making it an [[inner product space]]), and let <math display="block">W = \{\mathbf{u} \in V: \mathbf{A}x = \mathbf{u},\ x\in \R^2\},</math> with
<math display="block">\mathbf{A} = \begin{pmatrix}
1 & 0\\
0 & 1\\
2 & 6\\
3 & 9\\
5 & 3\\
\end{pmatrix}.</math>
then its orthogonal complement <math display="block">W^\perp = \{\mathbf{v}\in V:\langle \mathbf{u},\mathbf{v}\rangle = 0 \ \ \forall \ \mathbf{u} \in W\}</math> can also be defined as <math display="block">W^\perp = \{\mathbf{v} \in V: \mathbf{\tilde{A}}y = \mathbf{v},\ y \in \R^3\},</math> being
<math display="block">\mathbf{\tilde{A}} =
\begin{pmatrix}
-2 & -3 & -5 \\
-6 & -9 & -3 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{pmatrix}.</math>

The fact that every column vector in <math>\mathbf{A}</math> is orthogonal to every column vector in <math>\mathbf{\tilde{A}}</math> can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by bilinearity of the dot product. Finally, the fact that these spaces are orthogonal complements follows from the dimension relationships given below.

== General bilinear forms ==
Let <math>V</math> be a vector space over a [[Field (mathematics)|field]] <math>\mathbb{F}</math> equipped with a [[bilinear form]] <math>B.</math>  We define <math>\mathbf{u}</math> to be left-orthogonal to <math>\mathbf{v}</math>, and <math>\mathbf{v}</math> to be right-orthogonal to <math>\mathbf{u}</math>, when <math>B(\mathbf{u},\mathbf{v}) = 0.</math>  For a subset <math>W</math> of <math>V,</math> define the left-orthogonal complement <math>W^\perp</math> to be
<math display="block">W^\perp = \left\{ \mathbf{x} \in V : B(\mathbf{x}, \mathbf{y}) = 0 \ \ \forall \ \mathbf{y} \in W \right\}.</math>

There is a corresponding definition of the right-orthogonal complement. For a [[reflexive bilinear form]], where <math>B(\mathbf{u},\mathbf{v}) = 0 \implies B(\mathbf{v},\mathbf{u}) = 0 \ \ \forall \ \mathbf{u} , \mathbf{v} \in V</math>, the left and right complements coincide. This will be the case if <math>B</math> is a [[Symmetric bilinear form|symmetric]] or an [[Bilinear form#Symmetric, skew-symmetric and alternating forms|alternating form]].

The definition extends to a bilinear form on a [[free module]] over a [[commutative ring]], and to a [[sesquilinear form]] extended to include any free module over a commutative ring with [[Conjugate element (field theory)|conjugation]].<ref>Adkins & Weintraub (1992) p.359</ref>

=== Properties ===
* An orthogonal complement is a subspace of <math>V</math>;
* If <math>X \subseteq Y</math> then <math>X^\perp \supseteq Y^\perp</math>;
* The [[Radical of a quadratic space|radical]] <math>V^\perp</math> of <math>V</math> is a subspace of every orthogonal complement;
* <math>W \subseteq (W^\perp)^\perp</math>;
* If <math>B</math> is [[non-degenerate]] and <math>V</math> is finite-dimensional, then <math>\dim(W)+\dim (W^\perp)=\dim (V)</math>.
* If <math>L_1, \ldots, L_r</math> are subspaces of a finite-dimensional space <math>V</math> and <math>L_* = L_1 \cap \cdots \cap L_r,</math> then <math>L_*^\perp = L_1^\perp + \cdots + L_r^\perp</math>.

== Inner product spaces ==
{{See also|Orthogonal projection}}

This section considers orthogonal complements in an [[inner product space]] <math>H</math>.<ref>Adkins&Weintraub (1992) p.272</ref>

Two vectors <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are called {{em|[[Orthogonal vectors (inner product space)|orthogonal]]}} if <math>\langle \mathbf{x}, \mathbf{y} \rangle = 0</math>, which happens [[if and only if]] <math>\| \mathbf{x} \| \le \| \mathbf{x} + s\mathbf{y} \| \ \forall</math> scalars <math>s</math>.{{sfn|Rudin|1991|pp=306-312}}

If <math>C</math> is any subset of an inner product space <math>H</math> then its {{em|{{visible anchor|orthogonal complement}} in <math>H</math>}} is the vector subspace
<math display="block">\begin{align}
C^\perp
:&= \{\mathbf{x} \in H : \langle \mathbf{x}, \mathbf{c} \rangle = 0 \ \ \forall \ \mathbf{c} \in C\} \\
&= \{\mathbf{x} \in H : \langle \mathbf{c}, \mathbf{x} \rangle = 0 \ \ \forall \ \mathbf{c} \in C\} 
\end{align}</math>
which is always a closed subset (hence, a closed vector subspace) of <math>H</math>{{sfn|Rudin|1991|pp=306-312}}<ref group="proof">If <math>C = \varnothing</math> then <math>C^{\bot} = H,</math> which is closed in <math>H</math> so assume <math>C \neq \varnothing.</math> Let <math display="inline">P := \prod_{c \in C} \mathbb{F}</math> where <math>\mathbb{F}</math> is the underlying scalar field of <math>H</math> and define <math>L : H \to P</math> by <math>L(h) := \left(\langle h, c \rangle\right)_{c \in C},</math> which is continuous because this is true of each of its coordinates <math>h \mapsto \langle h, c \rangle.</math> Then <math>C^{\bot} = L^{-1}(0) = L^{-1}\left(\{ 0 \}\right)</math> is closed in <math>H</math> because <math>\{ 0 \}</math> is closed in <math>P</math> and <math>L : H \to P</math> is continuous. If <math>\langle \,\cdot\,, \,\cdot\, \rangle</math> is linear in its first (respectively, its second) coordinate then <math>L : H \to P</math> is a [[linear map]] (resp. an [[antilinear map]]); either way, its kernel <math>\operatorname{ker} L = L^{-1}(0) = C^{\bot}</math> is a vector subspace of <math>H.</math> [[Q.E.D.]]</ref> that satisfies:
* <math>C^{\bot} = \left(\operatorname{cl}_H \left(\operatorname{span} C\right)\right)^{\bot}</math>;
* <math>C^{\bot} \cap \operatorname{cl}_H \left(\operatorname{span} C\right) = \{ 0 \}</math>;
* <math>C^{\bot} \cap  \left(\operatorname{span} C\right) = \{ 0 \}</math>;
* <math>C \subseteq \left(C^{\bot}\right)^{\bot}</math>;
* <math>\operatorname{cl}_H \left(\operatorname{span} C\right) \subseteq \left(C^{\bot}\right)^{\bot}</math>.

If <math>C</math> is a vector subspace of an inner product space <math>H</math> then 
<math display="block">C^{\bot} = \left\{\mathbf{x} \in H : \|\mathbf{x}\| \leq \|\mathbf{x} + \mathbf{c}\| \ \ \forall \ \mathbf{c} \in C \right\}.</math> 
If <math>C</math> is a closed vector subspace of a Hilbert space <math>H</math> then{{sfn|Rudin|1991|pp=306-312}} 
<math display="block">H = C \oplus C^{\bot} \qquad \text{ and } \qquad \left(C^{\bot}\right)^{\bot} = C</math>
where <math>H = C \oplus C^{\bot}</math> is called the {{em|{{visible anchor|orthogonal decomposition}}}} of <math>H</math> into <math>C</math> and <math>C^{\bot}</math> and it indicates that <math>C</math> is a [[complemented subspace]] of <math>H</math> with complement <math>C^{\bot}.</math>

=== Properties ===

The orthogonal complement is always closed in the metric topology.  In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed.  In infinite-dimensional [[Hilbert space]]s, some subspaces are not closed, but all orthogonal complements are closed. If <math>W</math> is a vector subspace of a [[Hilbert space]] the orthogonal complement of the orthogonal complement of <math>W</math> is the [[Closure (topology)|closure]] of <math>W,</math> that is,
<math display="block">\left(W^\bot\right)^\bot = \overline W.</math>

Some other useful properties that always hold are the following. Let <math>H</math> be a Hilbert space and let <math>X</math> and <math>Y</math> be linear subspaces. Then:
* <math>X^\bot = \overline{X}^{\bot}</math>;
* if <math>Y \subseteq X</math> then <math>X^\bot \subseteq Y^\bot</math>;
* <math>X \cap X^\bot = \{ 0 \}</math>;
* <math>X \subseteq (X^\bot)^\bot</math>;
* if <math>X</math> is a closed linear subspace of <math>H</math> then <math>(X^\bot)^\bot = X</math>;
* if <math>X</math> is a closed linear subspace of <math>H</math> then <math>H = X \oplus X^\bot,</math> the (inner) [[direct sum]].

The orthogonal complement generalizes to the [[Annihilator (ring theory)|annihilator]], and gives a [[Galois connection]] on subsets of the inner product space, with associated [[closure operator]] the topological closure of the span.

=== Finite dimensions ===

For a finite-dimensional inner product space of dimension <math>n</math>, the orthogonal complement of a <math>k</math>-dimensional subspace is an <math>(n-k)</math>-dimensional subspace, and the double orthogonal complement is the original subspace:
<math display="block">\left(W^{\bot}\right)^{\bot} = W.</math>

If <math>\mathbf{A} \in \mathbb{M}_{mn}</math>, where <math>\mathcal{R}(\mathbf{A})</math>, <math>\mathcal{C} (\mathbf{A})</math>, and <math>\mathcal{N} (\mathbf{A})</math> refer to the [[row space]], [[column space]], and [[null space]] of <math>\mathbf{A}</math> (respectively), then<ref>[https://www.mathwizurd.com/linalg/2018/12/10/orthogonal-complement "Orthogonal Complement"]</ref>
<math display="block">\left(\mathcal{R} (\mathbf{A}) \right)^{\bot} = \mathcal{N} (\mathbf{A}) \qquad \text{ and } \qquad \left(\mathcal{C} (\mathbf{A}) \right)^{\bot} = \mathcal{N} (\mathbf{A}^{\operatorname{T}}).</math>

==Banach spaces==
There is a natural analog of this notion in general [[Banach space]]s. In this case one defines the orthogonal complement of <math>W</math> to be a subspace of the [[Dual space|dual]] of <math>V</math> defined similarly as the [[Dual space#Annihilators|annihilator]]
<math display="block">W^\bot = \left\{ x\in V^* : \forall y\in W, x(y) = 0 \right\}. </math>

It is always a closed subspace of <math>V^*</math>. There is also an analog of the double complement property. <math>W^{\perp \perp}</math> is now a subspace of <math>V^{**}</math>(which is not identical to <math>V</math>). However, the [[reflexive space]]s have a [[natural transformation|natural]] [[isomorphism]] <math>i</math> between <math>V</math> and <math>V^{**}</math>. In this case we have <math display="block">i\overline{W} = W^{\perp\perp}.</math>

This is a rather straightforward consequence of the [[Hahn–Banach theorem]].

==Applications==
In [[special relativity]] the orthogonal complement is used to determine the [[world line#Simultaneous hyperplane|simultaneous hyperplane]] at a point of a [[world line]]. The bilinear form <math>\eta</math> used in [[Minkowski space]] determines a [[pseudo-Euclidean space]] of events.<ref>[[G. D. Birkhoff]] (1923) ''Relativity and Modern Physics'', pages 62,63, [[Harvard University Press]]</ref> The origin and all events on the [[light cone]] are self-orthogonal. When a [[time]] event and a [[space]] event evaluate to zero under the bilinear form, then they are [[hyperbolic-orthogonal]]. This terminology stems from the use of [[conjugate hyperbola]]s in the pseudo-Euclidean plane: [[conjugate diameters]] of these hyperbolas are hyperbolic-orthogonal.

== See also ==

* {{annotated link|Complemented lattice}}
* {{annotated link|Complemented subspace}}
* {{annotated link|Hilbert projection theorem}}
* {{annotated link|Orthogonal projection}}

== Notes ==
{{reflist|group=proof}}

== References ==

{{reflist}}

== Bibliography ==

* {{citation|last1=Adkins|first1=William A.|last2=Weintraub|first2=Steven H.|title=Algebra: An Approach via Module Theory| series=[[Graduate Texts in Mathematics]]|volume=136|publisher=[[Springer-Verlag]]|year=1992|isbn=3-540-97839-9| zbl=0768.00003 }}
* {{Citation|last1=Halmos|first1=Paul R.|author1-link=Paul R. Halmos|title=Finite-dimensional vector spaces|series=[[Undergraduate Texts in Mathematics]]|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-0-387-90093-3| year=1974|zbl=0288.15002 }}
* {{citation|first1=J.|last1=Milnor|author1-link=John Milnor| first2=D.|last2=Husemoller|title=Symmetric Bilinear Forms|series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]]|volume=73|publisher=[[Springer-Verlag]]|year=1973|isbn=3-540-06009-X| zbl=0292.10016 }}
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->

== External links ==

* Orthogonal [https://www.youtube.com/watch?v=rnNxUekd3B4&t=645s complement; Minute 9.00 in the Youtube Video]
* [http://www.khanacademy.org/math/linear-algebra/alternate-bases/othogonal-complements/v/linear-algebra-orthogonal-complements Instructional video describing orthogonal complements (Khan Academy)]

{{Functional analysis}}
{{Hilbert space}}

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[[Category:Linear algebra]]
[[Category:Functional analysis]]