Quadratic formulas for quaternions

Conjugacy Classification of Quaternionic Möbius Transformations

“…In [3] Cao, Parker and Wang consider the subgroup of quaternionic Möbius transformations preserving the unit ball in H. Using the methods of Huang and So ( [9]) they calculate the fixed points of such quaternionic Möbius transformations. They classify these transformations as elliptic, parabolic or loxodromic and 1-simple or 2-simple using an invariant which, in our notation, is equal to γ − δ 2 − 2.…”

“…We review some basic linear algebra in SL(2, H) which can be found in [2,3,9,12,14], and relate the quantities α, β, γ, δ, σ and τ to eigenvalues of quaternionic matrices. We denote the real part of a quaternion z by Re [z].…”

“…Following Niven (see [3,9]), we combine this quadratic equation with (3.1) to obtain, provided τ = 2Re(t),…”

On the classification of quaternionic Möbius transformations

“…We review some basic linear algebra in SL(2, H) which can be found in [2,3,9,12,14], and relate the quantities α, β, γ, δ, σ and τ to eigenvalues of quaternionic matrices. We denote the real part of a quaternion z by Re A quaternion t is a right eigenvalue of a matrix A in SL(2, H) if and only if there is a non-zero column vector v ∈ H 2 such that Av = vt. For u = 0, let w = vu −1 .…”

On the Classification of Four-Dimensional Möbius Transformations

“…For a non-trivial element g, since the cardinality of its fixed point(s) and the norms of its right eigenvalues are conjugate invariant [5,6], the above classification is conjugate invariant and complete.…”