Mathematics-Online lexicon: Orthogonal and Special Orthogonal Group

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	Mathematics-Online lexicon:
	Orthogonal and Special Orthogonal Group

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The set of all orthogonal $(n\times n)$ -matrices

is called orthogonal group

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 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 2/10/2005