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.

Example edit

Let   be the vector space equipped with the usual dot product   (thus making it an inner product space), and let

then its orthogonal complement
can also be defined as

The fact that every column vector in   is orthogonal to every column vector in   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 edit

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   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 edit

  • 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  
  • If   are subspaces of a finite-dimensional space   and   then  

Inner product spaces edit

This section considers orthogonal complements in an inner product space  [2] Two vectors   and   are called orthogonal if   which happens if and only if   for all scalars  [3] If   is any subset of an inner product space   then its orthogonal complement in   is the vector subspace

which is always a closed subset of  [3][proof 1] that satisfies   and also   and   If   is a vector subspace of an inner product space   then
If   is a closed vector subspace of a Hilbert space   then[3]
where   is called the orthogonal decomposition of   into   and   and it indicates that   is a complemented subspace of   with complement  

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. If   is a vector subspace of an inner product space the orthogonal complement of the orthogonal complement of   is the closure of   that is,


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 edit

For a finite-dimensional inner product space of dimension   the orthogonal complement of a  -dimensional subspace is an  -dimensional subspace, and the double orthogonal complement is the original subspace:


If   is an   matrix, where     and   refer to the row space, column space, and null space of   (respectively), then[4]


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


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.

Applications 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.[5] 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.

See also edit

Notes edit

  1. ^ If   then   which is closed in   so assume   Let   where   is the underlying scalar field of   and define   by   which is continuous because this is true of each of its coordinates   Then   is closed in   because   is closed in   and   is continuous. If   is linear in its first (respectively, its second) coordinate then   is a linear map (resp. an antilinear map); either way, its kernel   is a vector subspace of   Q.E.D.

References edit

  1. ^ Adkins & Weintraub (1992) p.359
  2. ^ Adkins&Weintraub (1992) p.272
  3. ^ a b c Rudin 1991, pp. 306–312.
  4. ^ "Orthogonal Complement"
  5. ^ G. D. Birkhoff (1923) Relativity and Modern Physics, pages 62,63, Harvard University Press

Bibliography edit

  • Adkins, William A.; Weintraub, Steven H. (1992), Algebra: An Approach via Module Theory, Graduate Texts in Mathematics, vol. 136, Springer-Verlag, ISBN 3-540-97839-9, Zbl 0768.00003
  • Halmos, Paul R. (1974), Finite-dimensional vector spaces, Undergraduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90093-3, Zbl 0288.15002
  • Milnor, J.; Husemoller, D. (1973), Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, Springer-Verlag, ISBN 3-540-06009-X, Zbl 0292.10016
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.

External links edit

  • Orthogonal complement; Minute 9.00 in the Youtube Video
  • Instructional video describing orthogonal complements (Khan Academy)