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.
In the case that W is the subspace of (with the usual dot product) spanned by the rows of the next matrix,
its orthogonal complement W⊥ is spanned by the three row-vectors of
The fact that every vector on the first list is orthogonal to every vector on the second list 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
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 we 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.
- 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
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 is the closure of , i.e.,
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.
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.
- (Row A)⊥ = Null A
- (Col A)⊥ = Null AT.
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.
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.
- Adkins & Weintraub (1992) p.359
- Adkins&Weintraub (1992) p.272
- Adkins, William A.; Weintraub, Steven H. (1992), Algebra: An Approach via Module Theory, Graduate Texts in Mathematics, 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, 73, Springer-Verlag, ISBN 3-540-06009-X, Zbl 0292.10016