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

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...

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...

× n orthogonal matrices, under multiplication, forms the group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices...

transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n), for inhomogeneous special orthogonal group. The Euclidean...

symplectic form, and that this J is orthogonal; writing all the groups as matrix groups fixes a J (which is orthogonal) and ensures compatibility). In fact...

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...

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

orders of such groups, with a view to classifying cases of coincidence. A classical group is, roughly speaking, a special linear, orthogonal, symplectic...

conformal groups are particularly important: The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group...

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)...

neither simple nor semisimple. Another counter-example are the special orthogonal groups in even dimension. These have the matrix − I {\displaystyle -I} in...

whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space R...

space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set (or orthogonal system). If the vectors...

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

circle group is isomorphic to the special orthogonal group ⁠ S O ( 2 ) {\displaystyle \mathrm {SO} (2)} ⁠. One way to think about the circle group is that...

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

algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups, including orthogonal groups...

orthogonal group O(n, K), special orthogonal group SO(n, K), and symplectic group Sp(n, K)) are Lie groups that act on the vector space Kn. The group...

be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries...

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...

a right angle, whereas orthogonal is used in generalizations, such as orthogonal vectors or orthogonal curves. Orthogonality is also used with various...

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...

of the orthogonal group O ( n ) {\displaystyle O(n)} . It is a product of n orthogonal reflections (reflection through the axes of any orthogonal basis);...

2) is given by matrix multiplication: φ(h)(n) = hn. The orthogonal group O(n) of all orthogonal real n × n matrices (intuitively the set of all rotations...

multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M T M = I n {\displaystyle M^{T}M=I_{n}} where...

symmetry is isomorphic with the circle group in the complex plane, or the special orthogonal group SO(2), and unitary group U(1). Reflective circular symmetry...

group T and the torus groups Tn, the orthogonal group O(n), the special orthogonal group SO(n) and its covering spin group Spin(n), the unitary group...

the orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO(n)...

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...