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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| orthogonal group O(n) canonical | 3 |
| O(3) | 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.
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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.
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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.
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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.
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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) 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: orthogonal group O(n) Description of subject: 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.
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)