Orthogonal complement

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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[edit]

Let V be a vector space over a field F equipped with a bilinear form B. We define u to be left-orthogonal to v, and v to be right-orthogonal to u, when B(u,v)=0. For a subset W of V we define the left orthogonal complement W^\bot to be

W^\bot=\left\{x\in V : B( x, y ) = 0 \mbox{ for all } y\in W \right\}\,.

There is a corresponding definition of right orthogonal complement. For a reflexive bilinear form, where B(u,v)=0 implies B(v,u)=0 for all u and v in V, the left and right complements coincide. This will be the case if B 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[edit]

  • An orthogonal complement is a subspace of V;
  • If X\subset Y then X^\bot \supset Y^\bot;
  • The radical V^\bot of V is a subspace of every orthogonal complement;
  • W\subset (W^\bot)^\bot;
  • If B is non-degenerate and V is finite-dimensional, then \dim(W)+\dim (W^\bot)=\dim V.

Example[edit]

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[edit]

This section considers orthogonal complements in inner product spaces.[2]

Properties[edit]

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 W is the closure of W, i.e.,

(W^\bot)^\bot = \overline W .

Some other useful properties that always hold are the following. Let H be a Hilbert space and let X and Y be its linear subspaces. Then:

  • X^\bot = \overline X^\bot  ;
  • if  Y \subset X, then  X^\bot \subset Y ^\bot;
  • X\cap X^\bot = \{0\};
  • X\subset (X^\bot)^\bot  ;
  • if X is a closed linear subspace of H, then (X^\bot)^\bot = X;
  • if X is a closed linear subspace of H, then H = X \oplus X^\bot , 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[edit]

For a finite-dimensional inner product space of dimension n, the orthogonal complement of a k-dimensional subspace is an (nk)-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.

Banach spaces[edit]

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

W^\bot = \left\{\,x\in V^* : \forall y\in W, x(y) = 0 \, \right\}.\,

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

i\overline{W} = W^{\bot\,\bot}.

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

See also[edit]

References[edit]

  1. ^ Adkins & Weintraub (1992) p.359
  2. ^ Adkins&Weintraub (1992) p.272

External links[edit]