Orthogonality (mathematics)

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms.

Two elements u and v of a vector space with bilinear form

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.

In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface.

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 Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (

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.

[5] 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.

[5] By using integral calculus, it is common to use the following to define the inner product of two functions

with respect to a nonnegative weight function

are orthogonal if their inner product (equivalently, the value of this integral) is zero: Orthogonality of two functions with respect to one inner product does not imply orthogonality with respect to another inner product.

We write the norm with respect to this inner product as The members of a set of functions

if The members of such a set of functions are orthonormal with respect to

Various polynomial sequences named for mathematicians of the past are sequences of orthogonal polynomials.

Latin squares are said to be orthogonal if their superimposition yields all possible

of a Euclidean four-dimensional space are called completely orthogonal if and only if every line in

In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point.

Without loss of generality, we may take these to be the axes and orthogonal central planes of a

Each of the 6 orthogonal planes shares an axis with 4 of the others, and is completely orthogonal to just one of the others: the only one with which it does not share an axis.

Thus there are 3 pairs of completely orthogonal planes:

dimensions are called completely orthogonal if every line in