# Continuous Groups With Representations

• Albert Fässler
• Eduard Stiefel

## Abstract

We resume Example 1.5 where the group G of all proper rotations of the space R3 was introduced. After choosing a cartesian coordinate system (i.e., an orthonormal basis) in R3, we can describe such a rotation by the following matrix A: the kth basis vector is rotated into a vector a k with coordinates a1k, a2k, a3k Thus these coordinates must be written in the kth. column of A, thus
$$A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}{a_{12}}{a_{13}}} \\ {{a_{21}}{a_{22}}{a_{23}}} \\ {{a_{31}}{a_{32}}{a_{33}}} \end{array}} \right]$$
The three vectors 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)
Here 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.

Dinates Wolfram

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