We have a matrix C such that the map c(x) = Cx fixes Z(0,0,1), and has the property that C^{T}KC = K, where K = diag(1,1,-1).
Since c maps Z to Z, the third column of C must be (0,0,1)^{T}.
where 0 is the zero vector in R^{2},
u is some vector in R^{2}, and A direct calculation shows that
But this is K, so u = 0, and hence U^{T}U = I, so that U is orthogonal, |
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