mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Cartan–Dieudonné theorem, named after

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...

and Jean Dieudonné, establishes that every

orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we hav ...

in an ''n''- dimensional symmetric bilinear space can be described as the

composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

of at most ''n'' reflections. The notion of a symmetric bilinear space is a generalization of

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

whose structure is defined by a

symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a biline ...

(which need not be positive definite, so is not necessarily an

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

– for instance, a

pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space of signature is a finite- dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vect ...

is also a symmetric bilinear space). The orthogonal transformations in the space are those automorphisms which preserve the value of the bilinear form between every pair of vectors; in Euclidean space, this corresponds to preserving distances and

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

s. These orthogonal transformations form a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

under composition, called the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

. For example, in the two-dimensional Euclidean plane, every orthogonal transformation is either a reflection across a line through the origin or a

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

about the origin (which can be written as the composition of two reflections). Any arbitrary composition of such rotations and reflections can be rewritten as a composition of no more than 2 reflections. Similarly, in three-dimensional Euclidean space, every orthogonal transformation can be described as a single reflection, a rotation (2 reflections), or an

improper rotation In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection ( ...

(3 reflections). In four dimensions,

double rotation In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article ''rotation'' means ''rotational dis ...

s are added that represent 4 reflections.

Formal statement

Let be an ''n''-dimensional,

non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ' ...

symmetric bilinear space over a field with characteristic not equal to 2. Then, every element of the orthogonal group is a composition of at most ''n'' reflections.

References

* * * * Theorems in group theory Bilinear forms {{Abstract-algebra-stub

Formal statement

See also

References