In mathematics, a "'Clifford algebra "'is an algebra generated by a vector space with a quadratic form, and is a orthogonal transformations.
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Orthogonal transformations in two-or three-dimensional Euclidean space are stiff reflections, or combinations of a rotation and a reflection ( also known as improper rotations ).
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Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by-1, and can also be written as-I, where I is the identity matrix.
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Any element of E ( " n " ) is a translation followed by an orthogonal transformation ( the linear part of the isometry ), in a unique way:
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M�bius transformations can be more generally defined in spaces of dimension " n " > 2 as the bijective conformal orientation-preserving maps from the similarities, orthogonal transformations and inversions.
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Similarly, in three-dimensional Euclidean space, every orthogonal transformation can be described as a single reflection, a rotation ( 2 reflections ), or an improper rotation ( 3 reflections ).
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This is seen to preserve the Lagrangian, since the derivative of \ Phi transforms identically to \ Phi and both quantities appear inside dot products in the Lagrangian ( orthogonal transformations preserve the dot product ).
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PCA only relies on orthogonal transformations of the original data, and it only exploits the first-and second-order moments of the data, which may not well characterize the distribution of the data.
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Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate ( and simpler ) definition.
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Since " O " ( 4 ) acts on "'R "'4 by orthogonal transformations, we can move the " standard " Clifford torus defined above to other equivalent tori via rigid rotations.
