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.
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 we 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.
- 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 .
In special relativity the orthogonal complement is used to determine the simultaneous hyperplane at a point of a world line. The bilinear form η used in Minkowski space determines a pseudo-Euclidean space of events. 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 two conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal.
Inner product spaces
This section considers orthogonal complements in inner product spaces.
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. In such spaces, the orthogonal complement of the orthogonal complement of is the closure of , i.e.,
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.
For a finite-dimensional inner product space of dimension n, the orthogonal complement of a k-dimensional subspace is an (n − k)-dimensional subspace, and the double orthogonal complement is the original subspace:
- (W⊥)⊥ = W.
If A is an m × n matrix, where Row A, Col A, and Null A refer to the row space, column space, and null space of A (respectively), we have
- (Row A)⊥ = Null A
- (Col A)⊥ = Null AT.
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.