Math 2.0

Linear and multilinear algebra; matrix theory Matrix equations and groups Object

Orthogonal group

Builds on Orthogonal matrix

Description

Let n≥1n \ge 1. The orthogonal group is the set O(n)={Q∈Mn(ℝ):QTQ=In}O(n) = \{Q \in M_n(\R) : Q^T Q = I_n\} of real n×nn \times n orthogonal matrices, with matrix multiplication. It is a subgroup of the general linear group GLn(ℝ)GL_n(\R).

Remarks

It is the group of linear isometries of ℝn\R^n with the dot product. Each element has determinant 11 or −1-1, and det⁡ ⁣:O(n)→{1,−1}\det \colon O(n) \to \{1, -1\} is a surjective homomorphism. Its kernel is the special orthogonal group. O(n)O(n) is a compact Lie group of dimension n(n−1)/2n(n-1)/2.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionTranspose, definition, no evidence levelTransposedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobject
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  1. Identity matrix object
  2. Matrix definition
  3. Transpose definition
  4. Orthogonal matrix definition
  5. Orthogonal group object

Where this leads

38 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Orthogonal_group, lead section
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What this page needs for W2

  • section Invariants: a table of the main invariants (0 of 1)

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