orthogonal group O(n)
E518473
The orthogonal group O(n) is the group of all n×n real matrices that preserve the standard Euclidean inner product, representing rotations and reflections in n-dimensional space.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| O(3) | 1 |
| special orthogonal group SO(n) | 1 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
classical group
ⓘ
linear algebraic group ⓘ mathematical group ⓘ matrix group ⓘ |
| actsOn | n-dimensional real vector space R^n ⓘ |
| appearsIn |
differential geometry
ⓘ
physics ⓘ representation theory ⓘ |
| consistsOf |
linear isometries of R^n
ⓘ
n×n real matrices A with A^T A = I_n ⓘ |
| contains |
orthogonal transformations of R^n
ⓘ
reflections of R^n ⓘ rotations of R^n ⓘ |
| definedOver | real numbers ⓘ |
| hasCenter | {±I_n} for n even and {I_n} for n odd, for n ≥ 3 ⓘ |
| hasConnectedComponentOfIdentity | SO(n) NERFINISHED ⓘ |
| hasDeterminantCondition | det(A) = ±1 ⓘ |
| hasIdentityElement | identity matrix I_n ⓘ |
| hasIndex | 2 in O(n) for SO(n) ⓘ |
| hasLieAlgebra | skew-symmetric n×n real matrices ⓘ |
| hasOrder | infinite for all n ≥ 1 ⓘ |
| hasProperty |
Lie group of dimension n(n−1)/2
ⓘ
closed subgroup of GL(n,R) ⓘ closed under matrix multiplication ⓘ closed under taking inverses ⓘ compact ⓘ non-abelian for n ≥ 3 ⓘ |
| hasRank | floor(n/2) as a compact Lie group ⓘ |
| hasSubgroup | special orthogonal group SO(n) NERFINISHED ⓘ |
| hasTwoComponents | det(A) = 1 and det(A) = −1 ⓘ |
| isClosedIn | space of n×n real matrices with standard topology ⓘ |
| isCompactBecause | it is closed and bounded in R^{n^2} ⓘ |
| isConnected | false for n ≥ 1 ⓘ |
| isDefinedByEquation |
A A^T = I_n
ⓘ
A^T A = I_n ⓘ |
| isFiniteFor | no positive integer n ⓘ |
| isGeneratedBy | reflections in R^n ⓘ |
| isIsometryGroupOf | standard Euclidean space R^n fixing the origin ⓘ |
| isMaximalCompactSubgroupOf | GL(n,R) NERFINISHED ⓘ |
| isSubsetOf | general linear group GL(n,R) ⓘ |
| isSymmetryGroupOf | unit sphere S^{n−1} in R^n ⓘ |
| isUnionOf | SO(n) and the set of orthogonal matrices with determinant −1 ⓘ |
| LieAlgebraNotation | 𝔬(n) ⓘ |
| numberOfConnectedComponents | 2 for n ≥ 1 ⓘ |
| preserves |
Euclidean distance on R^n
ⓘ
Euclidean norm on R^n ⓘ standard Euclidean inner product on R^n ⓘ |
| symbol | O(n) NERFINISHED ⓘ |
| usedToModel | rotational symmetries in n-dimensional Euclidean space ⓘ |
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
special orthogonal group SO(n)
subject surface form:
SO(3)
this entity surface form:
O(3)