# Orthogonal group:O(2,R)

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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: