Orthogonal group

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In mathematics, the grade orthogonal group n on a body ${displaystyle scriptstyle mathbb {F} }$ designated as ${displaystyle scriptstyle {text{O}}(n,mathbb {F})}$ It's the group of orthogonal matrices. n for n with tickets ${displaystyle scriptstyle mathbb {F} }$ with the group operation given by the multiplication of matrices. This is a subgroup of the general linear group ${displaystyle scriptstyle {text{GL}}(n,mathbb {F})}$ .

Each orthogonal matrix has a determining 1 or -1. The matrices n for n orthogonals with determining 1 form a normal subgroup of ${displaystyle scriptstyle {text{O}}(n,mathbb {F})}$ known as the special orthogonal group ${displaystyle scriptstyle {text{SO}}(n,mathbb {F})}$ also known as rotational group. If the feature ${displaystyle scriptstyle mathbb {F} }$ It's 2, then. ${displaystyle scriptstyle {text{O}}(n,mathbb {F})}$ and ${displaystyle scriptstyle {text{SO}}(n,mathbb {F})}$ match; otherwise the index ${displaystyle scriptstyle {text{SO}}(n,mathbb {F})}$ in ${displaystyle scriptstyle {text{O}}(n,mathbb {F})}$ It's 2.

${displaystyle scriptstyle {text{O}}(n,mathbb {F})}$ and ${displaystyle scriptstyle {text{SO}}(n,mathbb {F})}$ are algebraic groups, because the condition that a matrix is orthogonal, that is to say that its own transposed is its reverse, can be expressed as a set of polynomial equations in the inputs of the matrix.

The real O(n) and SO(n) groups

When the mathematical body on which the orthonormal group is built is the body ${displaystyle scriptstyle mathbb {R} }$ of real numbers, orthogonal group ${displaystyle scriptstyle mathrm {O} (n,mathbb {R})}$ and the special orthogonal group ${displaystyle scriptstyle mathrm {SO} (n,mathbb {R}) subset mathrm {O} (n,mathbb {R})}$ often it is simply denoted by ${displaystyle scriptstyle mathrm {O} (n)}$ and ${displaystyle scriptstyle mathrm {SO} (n)}$ if there is no confusion possible. In that case, the groups ${displaystyle scriptstyle mathrm {O} (n)}$ and ${displaystyle scriptstyle mathrm {SO} (n)}$ are real, compact and dimensioned Lie groups n (n -1)/2. Plus topologically ${displaystyle scriptstyle mathrm {O} (n)}$ has two related components, being ${displaystyle scriptstyle mathrm {SO} (n)}$ one of these two related components, namely, the component containing the identity matrix.

The real special orthogonal and real orthogonal groups have simple geometric interpretations. O(n, R) is isomorphic to the group of isometries of Rⁿ which leave the origin fixed. SO(n, R) is isomorphic to the group of rotations of Rⁿ which leaves the origin fixed.

SO(2, R) is isomorphic (as a Lie group) to the circle S¹, consisting of all complex numbers of absolute value 1, with the multiplication of complex numbers as a group operation. This isomorphism sends the complex number exp(φi) = cos(φ) + isin(φ) to the orthogonal matrix:

${displaystyle {begin{bmatrix}cos(phi)&-operatorname {sen}(phi)\operatorname {sen}(phi)&cos(phi)end{bmatrix}}quad mapsto quad e^{iphi }}$

The group SO(3, R), understood as the set of rotations of 3-dimensional space, is of great importance in science and engineering. For a detailed description, see rotation group.

In terms of algebraic topology, for n > 2 the fundamental group of SO(n, R') is cyclic of order 2, and the spin group Spin(n) is its universal covering. For n = 2 the fundamental group is infinite cyclic and the universal coverage corresponds to the real line.

The Lie algebra associated with the Lie groups O(n, R) and SO(n, R) consists of the real anti-symmetric matrices n by n, with the Lie bracket given by the commutator. This Lie algebra is often denoted by or(n, R) or by so(n, R).

Properties

[ Lie groups] ${displaystyle {mbox{SO}}(n,mathbb {R})}$ and ${displaystyle {mbox{O}}(n,mathbb {R})}$ dimension ${displaystyle n(n-1)/2}$ .
The group ${displaystyle {mbox{O}}(n),}$ It's not related.
The group ${displaystyle {mbox{SO}}(2),}$ is related, although not simply related. Stop. n  2 ${displaystyle {mbox{SO}}(n),}$ has a fundamental cyclical group of order 2.

The complex O(n,C) and SO(n,C) groups

On the field C of complex numbers, O(n, C) and SO(n, C) are complex Lie groups of dimension n (n -1)/2 over C (meaning the dimension over R is twice that). O(n, C) has two connected components, and SO(n, C b>) is the connected component that contains the identity matrix. For n ≥ 2 these groups are non-compact.

Exactly as in the real case SO(n, C) is not simply connected. For n > 2 the fundamental group of SO(n, C) is cyclic of order 2 while the fundamental group of SO(2, C) is infinite cyclic.

The complex Lie algebra associated with O(n, C) and SO( n, C) consists of the complex anti-symmetric matrices n by n, with the Lie bracket given by the commutator.

Related Issues

rotational group, SO(3, R)
generalized orthogonal group
unitary group
Follow-up group

Data: Q1783179

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