Math 2.0

Linear and multilinear algebra; matrix theory Object

Special orthogonal group

Builds on Orthogonal groupDeterminant

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.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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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
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Transpose definition
  6. Linear map definition
  7. Orthogonal matrix definition
  8. Multilinear map definition
  9. Orthogonal group object
  10. Alternating multilinear map definition
  11. Determinant definition
  12. Special orthogonal group object

Where this leads

30 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthogonal_group, Special orthogonal group
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MSC15B10, 20G20
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