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.

General bilinear forms

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

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

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

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

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 ${\displaystyle W}$ is the closure of ${\displaystyle W}$, i.e.,

${\displaystyle (W^{\bot })^{\bot }={\overline {W}}}$.

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

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

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

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

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

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

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

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

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