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Wikipedia / Orthogonal complement

In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace of a vector space equipped with a bilinear form that are orthogonal. Informally, it is called the perp, short for perpendicular complement.

Example

be the vector space equipped with the usual dot product (thus making it an inner product space

The fact that every column vector 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

be a vector space over a field equipped with a bilinear form

There is a corresponding definition of the right-orthogonal complement. For a reflexive bilinear form 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

Inner product spaces

This section considers orthogonal complements in an inner product space.

are called , which happens if and only if.

then its which is always a closed subset that satisfies:

  • ;

  • ;

  • ;

  • ;

  • .

then is called the is a complemented subspace

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 is a vector subspace of a Hilbert space is the closure

be linear subspaces. 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.

Finite dimensions

refer to the row space, column space, and null space (respectively), then

Banach spaces

There is a natural analog of this notion in general Banach spaces to be a subspace of the dual defined similarly as the annihilator

). However, the reflexive spaces have a natural isomorphism

This is a rather straightforward consequence of the Hahn–Banach theorem.

Applications

In special relativity the orthogonal complement is used to determine the simultaneous hyperplane at a point of a world line 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 conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal.

See also

Notes

References

Bibliography