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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.

Contents

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 ⊥ = { x ∈ V : B ( x , y ) = 0  for all  y ∈ W } .

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.

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 .

    • Inner product spaces

    This section considers orthogonal complements in inner product spaces.

    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 AT.

    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 ⊥ = { x ∈ V ∗ : ∀ y ∈ W , x ( y ) = 0 } .

    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 W ¯ = W ⊥ ⊥ .

    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. 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.

    References

    Orthogonal complement Wikipedia
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