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 $A{A}^T$ is the identity matrix. Equivalently, it can be defined as:

$\{\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)\mid a^2+b^2=c^2+d^2=1,ac+bd=0\}$.

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

$\{\left(\begin{array}{cc}a& b\\ -b& a\\ \end{array}\right),\left(\begin{array}{cccc}a& b& b& -a\\ \end{array}\right)\mid a^2+b^2=1\}$.

The subgroup of matrices with determinant $1$ (i.e., the matrices with $ad-bc=1$) is the special orthogonal group $SO(2,\mathbb{R})$. 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:

• It is the 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.