In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V.
Example
Let be the vector space equipped with the usual dot product (thus making it an inner product space), and let with
then its orthogonal complement can also be defined as being
The fact that every column vector in is orthogonal to every column vector in 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 be a vector space over a field equipped with a bilinear form We define to be left-orthogonal to , and to be right-orthogonal to when For a subset of define the left orthogonal complement to be
There is a corresponding definition of right orthogonal complement. For a reflexive bilinear form, where implies for all and in the left and right complements coincide. This will be the case if is a symmetric or an 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 conjugation.
Properties
An orthogonal complement is a subspace of ; If then ; The radical of is a subspace of every orthogonal complement; ; If is non-degenerate and is finite-dimensional, then If are subspaces of a finite-dimensional space and then
Inner product spaces
This section considers orthogonal complements in an inner product space
Two vectors and are called if which happens if and only if for all scalars
If is any subset of an inner product space then its is the vector subspace
which is always a closed subset of that satisfies and also and
If is a vector subspace of an inner product space then
If is a closed vector subspace of a Hilbert space then
where is called the of into and and it indicates that is a complemented subspace of with complement
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 spaces, some subspaces are not closed, but all orthogonal complements are closed. If is a vector subspace of an inner product space the orthogonal complement of the orthogonal complement of is the closure of that is,
Some other useful properties that always hold are the following. Let be a Hilbert space and let and be its linear subspaces. Then: ; if then ; ; ; if is a closed linear subspace of then ; if is a closed linear subspace of then the (inner) direct sum.
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.
Finite dimensions
For a finite-dimensional inner product space of dimension the orthogonal complement of a -dimensional subspace is an -dimensional subspace, and the double orthogonal complement is the original subspace:
If is an matrix, where and refer to the row space, column space, and null space of (respectively), then
Banach spaces
There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complement of W to be a subspace of the dual of V defined similarly as the annihilator
It is always a closed subspace of V∗. There is also an analog of the double complement property. W⊥⊥ is now a subspace of V∗∗ (which is not identical to V). However, the reflexive spaces have a natural isomorphism i between V and V∗∗. In this case we have
This is a rather straightforward consequence of the Hahn–Banach theorem.