Orthogonality (mathematics)

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B {\displaystyle B} are orthogonal when B ( u , v ) = 0 {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0} . Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces, families of functions are used to form an orthogonal basis, such as in the contexts of orthogonal polynomials, orthogonal functions, and combinatorics.

Definitions In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e. they form a right angle. Two vectors u and v in an inner product space V {\displaystyle V} are orthogonal if their inner product ⟨ u , v ⟩ {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle } is zero. This relationship is denoted u ⊥ v {\displaystyle \mathbf {u} \perp \mathbf {v} } . A set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set (or orthogonal system). If the vectors are normalized, they form an orthonormal system. An orthogonal matrix is a matrix whose column vectors are orthonormal to each other. An orthonormal basis is a basis whose vectors are both orthogonal and normalized (they are unit vectors). A conformal linear transformation preserves angles and distance ratios, meaning that transforming orthogonal vectors by the same conformal linear transformation will keep those vectors orthogonal. Two vector subspaces A {\displaystyle A} and B {\displaystyle B} of an inner product space V {\displaystyle V} are called orthogonal subspaces if each vector in A {\displaystyle A} is orthogonal to each vector in B {\displaystyle B} . The largest subspace of V {\displaystyle V} that is orthogonal to a given subspace is its orthogonal complement. Given a module M {\displaystyle M} and its dual M ∗ {\displaystyle M^{*}} , an element m ′ {\displaystyle m'} of M ∗ {\displaystyle M^{*}} and an element m {\displaystyle m} of M {\displaystyle M} are orthogonal if their natural pairing is zero, i.e. ⟨ m ′ , m ⟩ = 0 {\displaystyle \langle m',m\rangle =0} . Two sets S ′ ⊆ M ∗ {\displaystyle S'\subseteq M^{*}} and S ⊆ M {\displaystyle S\subseteq M} are orthogonal if each element of S ′ {\displaystyle S'} is orthogonal to each element of S {\displaystyle S} . A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent. In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. For example, the y-axis is normal to the curve y = x 2 {\displaystyle y=x^{2}} at the origin. However, normal may also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in probability and statistics. A vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two vectors results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality. In the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ {\displaystyle \phi } .

Euclidean vector spaces In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° ( π 2 {\textstyle {\frac {\pi }{2}}} radians), or one of the vectors is zero. Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension. The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace. In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa. Note that the geometric concept of two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle. In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.

Orthogonal functions

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