In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave...

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension...

generalized unitary group as an algebraic group, one can take its points over various algebras. Analogous to the indefinite orthogonal groups, one can define...

mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R)...

transformations Mathematically, the Lorentz group may be described as the indefinite orthogonal group O(1, 3), the matrix Lie group that preserves the quadratic form...

view of the classical groups, the group of squeeze mappings is SO+(1,1), the identity component of the indefinite orthogonal group of 2×2 real matrices...

is the indefinite orthogonal group O(3,1), a Lie group. In other words, the Lorentz group is O(3,1). As presented in this article, any Lie groups mentioned...

fields. For example, for the orthogonal group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the...

applications: see proper time, for example. The rotations group of such space is the indefinite orthogonal group O(q), also denoted as O(k, n − k) without a reference...

space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation...

isometry group is a compact orthogonal group O(n). This stands in contrast with the case of isotropic forms, when the corresponding group, the indefinite orthogonal...

representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely...

the Minkowski space. The indefinite orthogonal group O(1,n), also called the (n+1)-dimensional Lorentz group, is the Lie group of real (n+1)×(n+1) matrices...

geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V,Q) on the associated...

group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps...

Then, every element of the orthogonal group O(V, b) is a composition of at most n reflections. Indefinite orthogonal group Coordinate rotations and reflections...

that (p, q) lies on the unit hyperbola. Such matrices form the indefinite orthogonal group O(1,1) with one-dimensional Lie algebra spanned by the anti-diagonal...

every indefinite orthogonal group as well. It equals the identity if and only if the characteristic is 2. It is the longest element of the Coxeter group of...

2 − y 2 = 1. {\displaystyle x^{2}-y^{2}=1.} In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial...

onto the orthogonal group. We define the special orthogonal group to be the image of Γ0. If K does not have characteristic 2 this is just the group of elements...

speculative idea that the fundamental symmetry group of spacetime is the indefinite orthogonal group SO(4,1), that of de Sitter space. In the standard...

matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column...

directions: the gauge group of (pseudo-)Riemannian geometry must be the indefinite orthogonal group O(s,r) in general, or the Lorentz group O(3,1) for space-time...

also be defined as the quotient O(1, n) / O(1, n − 1) of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space. Topologically...

the special orthogonal group SOn(R) and quotients, the projective orthogonal group POn(R), and the projective special orthogonal group PSOn(R). In characteristic...

to the indefinite orthogonal group SO+(1,2). It follows that SL(2, R) is isomorphic to the spin group Spin(2,1)+. Elements of the modular group PSL(2, Z)...

symplectic group, Sp(2n, R), on the phase space. Hamiltonian mechanics Metaplectic group Orthogonal group Paramodular group Projective unitary group Representations...

multiplication, the rotation group is isomorphic to the special orthogonal group SO(3). Improper rotations correspond to orthogonal matrices with determinant...

(for example, a pseudo-Riemannian metric, on the algebra of the indefinite orthogonal group so(p,q)). That is, structure constants are often written with...

them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend...