Orthogonal group:O(2,R)

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 member of family::orthogonal group over reals and hence of an member of family::orthogonal group.

Alternative definitions
This group can be defined in the following other ways:


 * It is the member of family::generalized dihedral group corresponding to the circle group, i.e., it is the semidirect product of the circle group by a group of order two acting by the inverse map.