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Mathematics-Online course: Linear Algebra - Analytic Geometry - Orthogonal Groups

Decomposition

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A rotation $ Q$ in $ \mathbb{R}^3$ can be composed of rotations about the coordinate axes:

$\displaystyle Q = D_z D_y D_x \,. $

(Authors: Höllig/Reble/Höfert)
The rotation can be determined by eliminating the elements $ q_{21}, q_{31}, q_{32}$ of $ Q$ step by step:
$\displaystyle P_z^{-1}Q$ $\displaystyle =$ $\displaystyle \left(\begin{array}{ccc} \star & \star & \star\\ 0 & \star & \star\\ \star & \star & \star\\ \end{array}\right)$  
$\displaystyle P_y^{-1}P_z Q$ $\displaystyle =$ $\displaystyle \left(\begin{array}{ccc} 1 & \star & \star\\ 0 & \star & \star\\ 0 & \star & \star\\ \end{array}\right)$  
$\displaystyle P_x^{-1}P_y^{-1}P_z Q$ $\displaystyle =$ $\displaystyle \left(\begin{array}{ccc} 1 & \star & \star\\ 0 & 1 & \star\\ 0 & 0 & \star\\ \end{array}\right) = R \,.$  

One used the fact, that an arbitrary unit vector $ (v_1, v_2)^{\operatorname t}$ can be mapped onto $ (1,0)^{\operatorname t}$ by a rotation. For $ \operatorname{det} R = 1$ it follows $ r_{33} =1$ an for the columns are normed $ R$ has to be the unit matrix. This leads to the wanted representation.
(Authors: Höllig/Reble/Höfert)
  automatically generated 4/21/2005