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

Classification and Special Transformations

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The set of all orthogonal $ (n\times n)$-matrices $ Q$ is called orthogonal group $ O(n)$.

For $ Q\in O(n)$

$\displaystyle Q^{-1} = Q^{\operatorname t}\,, \quad \vert\operatorname{det}\,Q\vert=1 $

holds, and each coordinate transormation

$\displaystyle x \to x^\prime = Qx $

preserves length and scalar product.

The set of all orthogonal $ (n\times n)$-matrices $ Q$ with $ \operatorname{det} Q=1$ is called rotation group or special orthogonal group

$\displaystyle SO(n) \subset O(n) \, . $

(Authors: Höllig/Reble/Höfert)
  automatically generated 4/21/2005