orthogonal group O(n)

E518473

classical group linear algebraic group mathematical group matrix group

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (5)

Label	Occurrences
orthogonal group O(n) canonical	5
orthogonal group	2
O(3)	1
O(n)/SO(n)	1
special orthogonal group SO(n)	1

How this entity was disambiguated

This entity first appeared as the object of triple T5425603 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: orthogonal group O(n)
Context triple: [Euclidean group, hasSubgroup, orthogonal group O(n)]

A. Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
B. Lorentz group
The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
C. rotation group SO(3)
The rotation group SO(3) is the group of all rotations in three-dimensional space, represented by 3×3 orthogonal matrices with determinant 1, and plays a central role in classical mechanics, quantum mechanics, and geometry.
D. Gaussian orthogonal ensemble
The Gaussian orthogonal ensemble is a fundamental random matrix ensemble of real symmetric matrices with Gaussian-distributed entries, central to the study of eigenvalue statistics and universality in random matrix theory.
E. Schmidt orthogonalization
Schmidt orthogonalization is a mathematical procedure, also known as the Gram–Schmidt process, that converts a set of linearly independent vectors into an orthonormal set spanning the same subspace.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: orthogonal group O(n)
Target entity description: 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.

A. Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
B. Lorentz group
The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
C. rotation group SO(3)
The rotation group SO(3) is the group of all rotations in three-dimensional space, represented by 3×3 orthogonal matrices with determinant 1, and plays a central role in classical mechanics, quantum mechanics, and geometry.
D. Gaussian orthogonal ensemble
The Gaussian orthogonal ensemble is a fundamental random matrix ensemble of real symmetric matrices with Gaussian-distributed entries, central to the study of eigenvalue statistics and universality in random matrix theory.
E. Schmidt orthogonalization
Schmidt orthogonalization is a mathematical procedure, also known as the Gram–Schmidt process, that converts a set of linearly independent vectors into an orthonormal set spanning the same subspace.
F. None of above. chosen

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) ⓘ linked to: special orthogonal group SO(n)
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) ⓘ
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) ⓘ linked to: general linear group GL(n,R)
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) ⓘ
usedToModel	rotational symmetries in n-dimensional Euclidean space ⓘ

How these facts were elicited