# Orthogonal complement

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*.

## Contents

## 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.^{[1]}

### 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 .

## Example

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.^{[2]}

### 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. 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.

### Finite dimensions

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*A*^{T}.

## 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.

## See also

## References

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