Orthogonal group:O(2,R): Difference between revisions

Latest revision as of 22:18, 24 August 2009

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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:

${(\begin{matrix} a & b \\ c & d \end{matrix}) ∣ a^{2} + b^{2} = c^{2} + d^{2} = 1, a c + b d = 0}$ .

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

${(\begin{matrix} a & b \\ - b & a \end{matrix}), (\begin{matrix} a & b & b & - a \end{matrix}) ∣ a^{2} + b^{2} = 1}$ .

The subgroup of matrices with determinant $1$ (i.e., the matrices with $a d - b c = 1$ ) is the special orthogonal group $S O (2, R)$ . 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:

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.

@@ Line 2: / Line 2: @@
 ==Definition==
+===Main definition===
 This group is defined as the group of <math>2 \times 2</math> matrices <math>A</math> with real entries such that <math>AA^T</math> is the identity matrix. Equivalently, it can be defined as:
@@ Line 9: / Line 11: @@
 In fact, there are only two possible forms of such matrices:
-<math>\left \{ \begin{pmatrix} a & b \\ -b & a \\\end{pmatrix}, \begin{\matrix} a & b & b & -a \\\end{pmatrix} \mid a^2 + b^2 = 1 \right \}</math>.
+<math>\left \{ \begin{pmatrix} a & b \\ -b & a \\\end{pmatrix}, \begin{pmatrix} a & b & b & -a \\\end{pmatrix} \mid a^2 + b^2 = 1 \right \}</math>.
 The subgroup of matrices with determinant <math>1</math> (i.e., the matrices with <math>ad - bc = 1</math>) is the special orthogonal group <math>SO(2,\R)</math>. 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.