Math 2.0

object

Special orthogonal group

Description

Let n≥1n \ge 1. The special orthogonal group is the set SO(n)={Q∈O(n):det⁡Q=1}SO(n) = \{Q \in O(n) : \det Q = 1\} of real orthogonal n×nn \times n matrices with determinant 11, with matrix multiplication. It is a normal subgroup of index 22 of the orthogonal group O(n)O(n).

Remarks

Its elements are the rotations of Rn\R^n about the origin. SO(2)SO(2) consists of the rotation matrices (cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}, and it is commutative. Each element of SO(3)SO(3) is a rotation about an axis through the origin. SO(n)SO(n) is a connected compact Lie group.

Explanations

L1 human

The special orthogonal group contains the orthogonal matrices that do not flip orientation. These are the rotations. Reflections have determinant −1-1, so they are not in this group.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…object
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  1. Identity matrix object
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  8. Multilinear map definition
  9. Orthogonal group object
  10. Alternating multilinear map definition
  11. Determinant definition
  12. Special orthogonal group object

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Sources

  1. Web https://en.wikipedia.org/wiki/Orthogonal_group, Special orthogonal group

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