In themathematical fields oflinear algebra andfunctional analysis, theorthogonal complement of asubspace${\displaystyle W}$ of avector space${\displaystyle V}$ equipped with abilinear form${\displaystyle B}$ is the set${\displaystyle W^{\perp }}$ of all vectors in${\displaystyle V}$ that areorthogonal to every vector in${\displaystyle W}$. Informally, it is called theperp, short forperpendicular complement. It is a subspace of${\displaystyle V}$.

Let${\displaystyle V=({\mathbb{R}}^5,⟨\cdot ,\cdot ⟩)}$ be the vector space equipped with the usualdot product${\displaystyle ⟨\cdot ,\cdot ⟩}$ (thus making it aninner product space), and let$${\displaystyle W=\{\mathbf{u}\in V:\mathbf{A}x=\mathbf{u},x\in {\mathbb{R}}^2\},}$$ withthen its orthogonal complement$${\displaystyle W^{\perp }=\{\mathbf{v}\in V:⟨\mathbf{u},\mathbf{v}⟩=0\mathrm{\forall }\mathbf{u}\in W\}}$$ can also be defined as$${\displaystyle W^{\perp }=\{\mathbf{v}\in V:\stackrel{\mathbf{~}}{\mathbf{A}}y=\mathbf{v},y\in {\mathbb{R}}^3\},}$$ being

The fact that everycolumn vector in${\displaystyle \mathbf{A}}$ is orthogonal to every column vector in${\displaystyle \stackrel{\mathbf{~}}{\mathbf{A}}}$ 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.

Let${\displaystyle V}$ be a vector space over afield${\displaystyle \mathbb{F}}$ equipped with abilinear form${\displaystyle B.}$ We define${\displaystyle \mathbf{u}}$ to be left-orthogonal to${\displaystyle \mathbf{v}}$, and${\displaystyle \mathbf{v}}$ to be right-orthogonal to${\displaystyle \mathbf{u}}$, when${\displaystyle B(\mathbf{u},\mathbf{v})=0.}$ For a subset${\displaystyle W}$ of${\displaystyle V,}$ define the left-orthogonal complement${\displaystyle W^{\perp }}$ to be$${\displaystyle W^{\perp }=\left\{\mathbf{x}\in V:B(\mathbf{x},\mathbf{y})=0\mathrm{\forall }\mathbf{y}\in W\right\}.}$$

There is a corresponding definition of the right-orthogonal complement. For areflexive bilinear form, where${\displaystyle B(\mathbf{u},\mathbf{v})=0⟹B(\mathbf{v},\mathbf{u})=0\mathrm{\forall }\mathbf{u},\mathbf{v}\in V}$, the left and right complements coincide. This will be the case if${\displaystyle B}$ is asymmetric or analternating form.

The definition extends to a bilinear form on afree module over acommutative ring, and to asesquilinear form extended to include any free module over a commutative ring withconjugation.^[1]

• An orthogonal complement is a subspace of${\displaystyle V}$;
• If${\displaystyle X\subseteq Y}$ then${\displaystyle X^{\perp }\supseteq Y^{\perp }}$;
• Theradical${\displaystyle V^{\perp }}$ of${\displaystyle V}$ is a subspace of every orthogonal complement;
• ${\displaystyle W\subseteq (W^{\perp }{)}^{\perp }}$;
• If${\displaystyle B}$ isnon-degenerate and${\displaystyle V}$ is finite-dimensional, then${\displaystyle \dim (W)+\dim (W^{\perp })=\dim (V)}$.
• If${\displaystyle L_1,\dots ,L_r}$ are subspaces of a finite-dimensional space${\displaystyle V}$ and${\displaystyle L_{\ast }=L_1\cap \cdots \cap L_r,}$ then${\displaystyle L_{\ast }^{\perp }=L_1^{\perp }+\cdots +L_r^{\perp }}$.

This section considers orthogonal complements in aninner product space${\displaystyle H}$.^[2]

Two vectors${\displaystyle \mathbf{x}}$ and${\displaystyle \mathbf{y}}$ are calledorthogonal if${\displaystyle ⟨\mathbf{x},\mathbf{y}⟩=0}$, which happensif and only if${\displaystyle \Vert \mathbf{x}\Vert \le \Vert \mathbf{x}+s\mathbf{y}\Vert \mathrm{\forall }}$ scalars${\displaystyle s}$.^[3]

If${\displaystyle C}$ is any subset of an inner product space${\displaystyle H}$ then itsorthogonal complement in${\displaystyle H}$ is the vector subspacewhich is always aclosed subset (hence, a closed vector subspace) of${\displaystyle H}$^{[3][proof 1]} that satisfies:

• ${\displaystyle C^{\mathrm{\perp }}={\left({\mathrm{cl}}_H\left(\mathrm{span}C\right)\right)}^{\mathrm{\perp }}}$;
• ${\displaystyle C^{\mathrm{\perp }}\cap {\mathrm{cl}}_H\left(\mathrm{span}C\right)=\{0\}}$;
• ${\displaystyle C^{\mathrm{\perp }}\cap \left(\mathrm{span}C\right)=\{0\}}$;
• ${\displaystyle C\subseteq {\left(C^{\mathrm{\perp }}\right)}^{\mathrm{\perp }}}$;
• ${\displaystyle {\mathrm{cl}}_H\left(\mathrm{span}C\right)\subseteq {\left(C^{\mathrm{\perp }}\right)}^{\mathrm{\perp }}}$.

If${\displaystyle C}$ is a vector subspace of an inner product space${\displaystyle H}$ then$${\displaystyle C^{\mathrm{\perp }}=\left\{\mathbf{x}\in H:\Vert \mathbf{x}\Vert \le \Vert \mathbf{x}+\mathbf{c}\Vert \mathrm{\forall }\mathbf{c}\in C\right\}.}$$ If${\displaystyle C}$ is a closed vector subspace of a Hilbert space${\displaystyle H}$ then^[3]$${\displaystyle H=C\oplus C^{\mathrm{\perp }}\text{ and }{\left(C^{\mathrm{\perp }}\right)}^{\mathrm{\perp }}=C}$$where${\displaystyle H=C\oplus C^{\mathrm{\perp }}}$ is called theorthogonal decomposition of${\displaystyle H}$ into${\displaystyle C}$ and${\displaystyle C^{\mathrm{\perp }}}$ and it indicates that${\displaystyle C}$ is acomplemented subspace of${\displaystyle H}$ with complement${\displaystyle C^{\mathrm{\perp }}.}$

The orthogonal complement is always closed in themetric topology. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensionalHilbert spaces, some subspaces are not closed, but all orthogonal complements are closed. If${\displaystyle W}$ is a vector subspace of aHilbert space the orthogonal complement of the orthogonal complement of${\displaystyle W}$ is theclosure of${\displaystyle W,}$ that is,$${\displaystyle {\left(W^{\mathrm{\perp }}\right)}^{\mathrm{\perp }}=\overline{W}.}$$

Some other useful properties that always hold are the following. Let${\displaystyle H}$ be a Hilbert space and let${\displaystyle X}$ and${\displaystyle Y}$ be linear subspaces. Then:

• ${\displaystyle X^{\mathrm{\perp }}={\overline{X}}^{\mathrm{\perp }}}$;
• if${\displaystyle Y\subseteq X}$ then${\displaystyle X^{\mathrm{\perp }}\subseteq Y^{\mathrm{\perp }}}$;
• ${\displaystyle X\cap X^{\mathrm{\perp }}=\{0\}}$;
• ${\displaystyle X\subseteq (X^{\mathrm{\perp }}{)}^{\mathrm{\perp }}}$;
• if${\displaystyle X}$ is a closed linear subspace of${\displaystyle H}$ then${\displaystyle (X^{\mathrm{\perp }}{)}^{\mathrm{\perp }}=X}$;
• if${\displaystyle X}$ is a closed linear subspace of${\displaystyle H}$ then${\displaystyle H=X\oplus X^{\mathrm{\perp }},}$ the (inner)direct sum.

The orthogonal complement generalizes to theannihilator, and gives aGalois connection on subsets of the inner product space, with associatedclosure operator the topological closure of the span.

For a finite-dimensional inner product space of dimension${\displaystyle n}$, the orthogonal complement of a${\displaystyle k}$-dimensional subspace is an${\displaystyle (n-k)}$-dimensional subspace, and the double orthogonal complement is the original subspace:$${\displaystyle {\left(W^{\mathrm{\perp }}\right)}^{\mathrm{\perp }}=W.}$$

If${\displaystyle \mathbf{A}\in {\mathbb{M}}_{mn}}$, where${\displaystyle \mathcal{R}(\mathbf{A})}$,${\displaystyle \mathcal{C}(\mathbf{A})}$, and${\displaystyle \mathcal{N}(\mathbf{A})}$ refer to therow space,column space, andnull space of${\displaystyle \mathbf{A}}$ (respectively), then^[4]$${\displaystyle {\left(\mathcal{R}(\mathbf{A})\right)}^{\mathrm{\perp }}=\mathcal{N}(\mathbf{A})\text{ and }{\left(\mathcal{C}(\mathbf{A})\right)}^{\mathrm{\perp }}=\mathcal{N}({\mathbf{A}}^{\mathrm{T}}).}$$

There is a natural analog of this notion in generalBanach spaces. In this case one defines the orthogonal complement of${\displaystyle W}$ to be a subspace of thedual of${\displaystyle V}$ defined similarly as theannihilator$${\displaystyle W^{\mathrm{\perp }}=\left\{x\in V^{\ast }:\mathrm{\forall }y\in W,x(y)=0\right\}.}$$

It is always a closed subspace of${\displaystyle V^{\ast }}$. There is also an analog of the double complement property.${\displaystyle W^{\perp \perp }}$ is now a subspace of${\displaystyle V^{\ast \ast }}$(which is not identical to${\displaystyle V}$). However, thereflexive spaces have anaturalisomorphism${\displaystyle i}$ between${\displaystyle V}$ and${\displaystyle V^{\ast \ast }}$. In this case we have$${\displaystyle i\overline{W}=W^{\perp \perp }.}$$

This is a rather straightforward consequence of theHahn–Banach theorem.

Inspecial relativity the orthogonal complement is used to determine thesimultaneous hyperplane at a point of aworld line. The bilinear form${\displaystyle \eta }$ used inMinkowski space determines apseudo-Euclidean space of events.^[5] The origin and all events on thelight cone are self-orthogonal. When atime event and aspace event evaluate to zero under the bilinear form, then they arehyperbolic-orthogonal. This terminology stems from the use ofconjugate hyperbolas in the pseudo-Euclidean plane:conjugate diameters of these hyperbolas are hyperbolic-orthogonal.

• Complemented lattice – Bound lattice in which every element has a complement
• Complemented subspace – Concept in functional analysis
• Hilbert projection theorem – On closed convex subsets in Hilbert space
• Orthogonal projection – Idempotent linear transformation from a vector space to itselfPages displaying short descriptions of redirect targets

• Adkins, William A.; Weintraub, Steven H. (1992),Algebra: An Approach via Module Theory,Graduate Texts in Mathematics, vol. 136,Springer-Verlag,ISBN 3-540-97839-9,Zbl 0768.00003
• Halmos, Paul R. (1974),Finite-dimensional vector spaces,Undergraduate Texts in Mathematics, Berlin, New York:Springer-Verlag,ISBN 978-0-387-90093-3,Zbl 0288.15002
• Milnor, J.; Husemoller, D. (1973),Symmetric Bilinear Forms,Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73,Springer-Verlag,ISBN 3-540-06009-X,Zbl 0292.10016
• Rudin, Walter (1991).Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN 978-0-07-054236-5.OCLC 21163277.

• Orthogonalcomplement; Minute 9.00 in the Youtube Video
• Instructional video describing orthogonal complements (Khan Academy)

• Linear algebra
• Functional analysis

• Articles with short description
• Short description matches Wikidata
• Pages displaying short descriptions of redirect targets via Module:Annotated link