Orthogonal group:O(2,R)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

Main definition

This group is defined as the group of 2 \times 2 matrices A with real entries such that AA^T is the identity matrix. Equivalently, it can be defined as:

\left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a^2 + b^2 = c^2 + d^2 = 1, ac + bd = 0 \right \}.

In fact, there are only two possible forms of such matrices:

\left \{ \begin{pmatrix} a & b \\ -b & a \\\end{pmatrix}, \begin{pmatrix} a & b & b & -a \\\end{pmatrix} \mid a^2 + b^2 = 1 \right \}.

The subgroup of matrices with determinant 1 (i.e., the matrices with ad - bc = 1) is the special orthogonal group SO(2,\R). It has index two and is isomorphic to the circle group.

This group is a particular case of an orthogonal group over reals and hence of an orthogonal group.

Alternative definitions

This group can be defined in the following other ways: