# Continuous Groups With Representations

Chapter

## Abstract

We resume Example 1.5 where the group
The three vectors
Here

*G*of all proper rotations of the space**R**^{3}was introduced. After choosing a cartesian coordinate system (i.e., an orthonormal basis) in**R**^{3}, we can describe such a rotation by the following matrix*A*: the*k*th basis vector is rotated into a vector*a*_{ k }with coordinates*a*_{1k},*a*_{2k},*a*_{3k}Thus these coordinates must be written in the*k*th. column of*A*, thus$$
A = \left[ {\begin{array}{*{20}{c}}
{{a_{11}}{a_{12}}{a_{13}}} \\
{{a_{21}}{a_{22}}{a_{23}}} \\
{{a_{31}}{a_{32}}{a_{33}}}
\end{array}} \right]
$$

*a*_{ k }are orthogonal unit vectors. Therefore, the inner product of two distinct columns = 0, and the inner product of a column with itself = 1. Inshort,$$
{A^{\text{T}}}A = E{\text{ or equivalently }}{A^{ - 1}} = {A^T}
$$

(4.1)

*A*^{ T }is the transposed matix and*E*the 3×3 identity matrix. Real matrices with the property (4.1) are called**orthogonal**. Conxequently, the inverse of an orthogonal matrix is obtained by interchanging rown and columns.## Keywords

Irreducible Representation Homogeneous Polynomial Lorentz Group Abelian Subgroup Spherical Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1992