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- The invariant decomposition is a decomposition of the elements of groups into orthogonal commuting elements. It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition of oriented reflections, the invariant decomposition theorem reads Every -reflection can be decomposed into commuting factors. It is named the invariant decomposition because these factors are the invariants of the -reflection . A well known special case is the Chasles' theorem, which states that any rigid body motion in can be decomposed into a rotation around, followed or preceded by a translation along, a single line. Both the rotation and the translation leave two lines invariant: the axis of rotation and the orthogonal axis of translation. Since both rotations and translations are bireflections, a more abstract statement of the theorem reads "Every quadreflection can be decomposed into commuting bireflections". In this form the statement is also valid for e.g. the spacetime algebra , where any Lorentz transformation can be decomposed into a commuting rotation and boost. (en)
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- The invariant decomposition is a decomposition of the elements of groups into orthogonal commuting elements. It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition of oriented reflections, the invariant decomposition theorem reads Every -reflection can be decomposed into commuting factors. (en)
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- Invariant decomposition (en)
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