In
mathematics, 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 ...
and
Jean Dieudonné, establishes that every
orthogonal transformation in an ''n''-
dimensional symmetric bilinear space can be described as the
composition 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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
whose structure is defined by a
symmetric bilinear form (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, often ...
– for instance, a
pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space 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 vector , giving
q(x ...
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 is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
s. These orthogonal transformations form a
group under composition, called the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
.
For example, in the two-dimensional
Euclidean plane, every orthogonal transformation is either a reflection across a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
through the origin or a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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 about an axis and a reflection in a plane perpendic ...
(3 reflections). In four dimensions,
double rotations are added that represent 4 reflections.
Formal statement
Let be an ''n''-dimensional,
non-degenerate 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.
See also
*
Indefinite orthogonal group
*
Coordinate rotations and reflections
References
*
*
*
*
Theorems in group theory
Bilinear forms
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