In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace
equipped with a bilinear form
Informally, it is called the perp, short for perpendicular complement.
be the vector space equipped with the usual dot product
(thus making it an inner product space), and let
is orthogonal to every column vector in
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.
be a vector space over a field
equipped with a bilinear form
There is a corresponding definition of the right-orthogonal complement.
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 conjugation.
[1] This section considers orthogonal complements in an inner product space
is any subset of an inner product space
is a vector subspace of an inner product space
is a closed vector subspace of a Hilbert space
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 spaces, some subspaces are not closed, but all orthogonal complements are closed.
is a vector subspace of a Hilbert space the orthogonal complement of the orthogonal complement of
Then: The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span.
For a finite-dimensional inner product space of dimension
There is a natural analog of this notion in general Banach spaces.
In this case one defines the orthogonal complement of
There is also an analog of the double complement property.
However, the reflexive spaces have a natural isomorphism
In special relativity the orthogonal complement is used to determine the simultaneous hyperplane at a point of a world line.
[5] 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 hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal.