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 are orthogonal when . Depending on the bilinear form, the vector space may contain non-zero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form an orthogonal basis.
The concept has been used in the context of orthogonal functions, orthogonal polynomials, and combinatorics.
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.
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 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 .
In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° ( radians), or one of the vectors is zero. [4] 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. [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 and with respect to a nonnegative weight function over an interval :
In simple cases, .
We say that functions and 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 are orthogonal with respect to on the interval if
The members of such a set of functions are orthonormal with respect to on the interval if
where
is the Kronecker delta.
In other words, every pair of them (excluding pairing of a function with itself) is orthogonal, and the norm of each is 1. See in particular the orthogonal polynomials.
Various polynomial sequences named for mathematicians of the past are sequences of orthogonal polynomials. In particular:
In combinatorics, two Latin squares are said to be orthogonal if their superimposition yields all possible combinations of entries. [6]
Two flat planes and of a Euclidean four-dimensional space are called completely orthogonal if and only if every line in is orthogonal to every line in . [7] In that case the planes and intersect at a single point , so that if a line in intersects with a line in , they intersect at . and are perpendicular and Clifford parallel.
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 Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes that we have in 3 dimensions, and also 3 others . 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: and intersect only at the origin; and intersect only at the origin; and intersect only at the origin.
More generally, two flat subspaces and of dimensions and of a Euclidean space of at least dimensions are called completely orthogonal if every line in is orthogonal to every line in . If then and intersect at a single point . If then and may or may not intersect. If then a line in and a line in may or may not intersect; if they intersect then they intersect at . [8]
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 are orthogonal when . Depending on the bilinear form, the vector space may contain non-zero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form an orthogonal basis.
The concept has been used in the context of orthogonal functions, orthogonal polynomials, and combinatorics.
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.
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 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 .
In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° ( radians), or one of the vectors is zero. [4] 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. [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 and with respect to a nonnegative weight function over an interval :
In simple cases, .
We say that functions and 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 are orthogonal with respect to on the interval if
The members of such a set of functions are orthonormal with respect to on the interval if
where
is the Kronecker delta.
In other words, every pair of them (excluding pairing of a function with itself) is orthogonal, and the norm of each is 1. See in particular the orthogonal polynomials.
Various polynomial sequences named for mathematicians of the past are sequences of orthogonal polynomials. In particular:
In combinatorics, two Latin squares are said to be orthogonal if their superimposition yields all possible combinations of entries. [6]
Two flat planes and of a Euclidean four-dimensional space are called completely orthogonal if and only if every line in is orthogonal to every line in . [7] In that case the planes and intersect at a single point , so that if a line in intersects with a line in , they intersect at . and are perpendicular and Clifford parallel.
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 Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes that we have in 3 dimensions, and also 3 others . 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: and intersect only at the origin; and intersect only at the origin; and intersect only at the origin.
More generally, two flat subspaces and of dimensions and of a Euclidean space of at least dimensions are called completely orthogonal if every line in is orthogonal to every line in . If then and intersect at a single point . If then and may or may not intersect. If then a line in and a line in may or may not intersect; if they intersect then they intersect at . [8]