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...

× 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...

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)...

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...

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...

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...

is perpendicular to line B"), orthogonal is commonly used without to (e.g., "orthogonal lines A and B"). Orthogonality is also used with various meanings...

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...

The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO(n), one example...

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...

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...

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

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...

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

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)...

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...

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

is the orthogonal group O(n), while "the" maximal compact subgroup of GL+(n, R) is the special orthogonal group SO(n). As for SO(n), the group GL+(n,...

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

decomposition Orthogonal group O(n) Special orthogonal group SO(n) Orthogonal matrix Semi-orthogonal matrix Quantum logic gate Special Unitary group SU(n) Symplectic...

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);...

representing an element of S O ( n ) {\displaystyle SO(n)} (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix A T = −...

}. The center of the orthogonal group, On(F) is {In, −In}. The center of the special orthogonal group, SO(n) is the whole group when n = 2, and otherwise...

The orthogonal group is compact as a topological space. Much of Euclidean geometry can be viewed as studying the structure of the orthogonal group, or...

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

and the Euclidean group is the semidirect product of the translation group and the orthogonal group. The special orthogonal group is the normal subgroup...

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms. Two elements...