Cayley-Klein Parameters

Author

Eric W. Weisstein
March 23, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialGeometry/Cayley-KleinParameters.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Cayley-KleinParameters.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Identities

CayleyKlein={
α->Exp[I(ψ+ϕ)/2]Cos[(θ/2)],
β->I Exp[I(ψ-ϕ)/2]Sin[(θ/2)],
γ->I Exp[-I(ψ-ϕ)/2]Sin[(θ/2)],
δ->Exp[-I(ψ+ϕ)/2]Cos[(θ/2)]};

Simplify[ComplexExpand[{
((Norm[α])^2)+((Norm[γ])^2),
((Norm[α])^2)+((Norm[β])^2),
((Norm[β])^2)+((Norm[δ])^2),
Conjugate[α]β+Conjugate[γ]δ,
α δ-β γ,
β+Conjugate[γ],
δ-Conjugate[α]
}/.CayleyKlein
],0<ϕ<π&&0<θ<π&&0<ψ<π]

MatrixForm/@With[{Q={{α,β},{γ,δ}}/.CayleyKlein},
{
ComplexExpand[ConjugateTranspose[Q].Q],
ComplexExpand[Q.ConjugateTranspose[Q]]
}
]//Simplify