Orthogonal complement

/div>

Jump to: navigation, search

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.

1 General bilinear forms
- 1.1 Properties
2 Example
3 Inner product spaces
4 See also
5 References
6 External links

[ General bilinear forms

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^⊥ 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 skew-symmetric bilinear 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 V;

If X ⊆ Y then X^⊥ ⊇ Y^⊥;

The radical V^⊥ of V is a subspace of every orthogonal complement;
W^⊥^⊥ ⊇ W;
If B is non-degenerate and V is finite-dimensional, then dim W + dim W^⊥ = dim V.

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

(W^⊥)^⊥ = 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^⊥ = X^⊥;
if Y ⊆ X then X^⊥ ⊆ Y^⊥;
X ∩ X^⊥ = {0};
X ⊆ (X^⊥)^⊥;
if X is a closed linear subspace of H, then (X^⊥)^⊥ = X;
if X is a closed linear subspace of H, then H = X ⊕ X^⊥, 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

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

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