The Rotation Group and its Representations

Abstract

We saw in Example 2 of Section 3.8 that an operator R rotating a rectangular Cartesian system (ijk) of unit vectors into another system (i′j′k′) according to the transformation

$$\left( {\begin{array}{*{20}{c}}{i'} \\{j'} \\{k'} \\\end{array} } \right) = \left( {\begin{array}{*{20}{c}}{{l_1}} & {{l_2}} & {{l_3}} \\{{m_1}} & {{m_2}} & {{m_3}} \\{{n_1}} & {{n_2}} & {{n_3}} \\\end{array} } \right)\left( {\begin{array}{*{20}{c}}i \\j \\k \\\end{array} } \right) = T\left( {\begin{array}{*{20}{c}}i \\j \\k \\\end{array} } \right)$$

((8.1.1))

is represented relative to the basis (ijk) by the matrix

$$R = \tilde T = \left( {\begin{array}{*{20}{c}} {{l_1}} & {{m_1}} & {{n_1}} \\ {{l_2}} & {{m_2}} & {{n_2}} \\ {{l_3}} & {{m_3}} & {{n_3}} \\ \end{array} } \right)$$

((8.1.2))

which is proper orthogonal i.e., $\tilde RR = R\tilde R = E$, det R = +1. Observe that R is the transpose ${\tilde T}$ of the matrix T effecting a change of basis from (ijk) to (i′j′k′). We also note that a rotation preserves the scalar product of two vectors; for, if x′ = Rx and y′ = Ry, we have

$$ {\tilde{x}}^{\prime}{y}^{\prime}={{\left( {Rx} \right)}^{\sim }}\left( {Ry} \right)=\tilde{x}\tilde{R}Ry=\widetilde{{xy}}\quad \text{since}\quad \tilde{R}R=E. $$

.