By Augustus J.E.M. Janssen, P. van der Steen
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Example text
Diese gestattet dann die Darstellung x = z1 + z2 j. Eine Quaternion x ∈ S 3 entspricht dabei einem Paar komplexer Zahlen (z1 , z2 ) ∈ C× C = C2 mit |x|2 = |z1 |2 + |z2 |2 = 1. Auf natürliche Weise finden wir nun eine Abbildung m, die x eine unitäre Matrix vom Typ 2 × 2 mit komplexen Einträgen zuordnet. Wir definieren: m(x) := z1 −z2 z2 z1 ∈ C2×2 . Alle unitären Matrizen mit det m(x) = 1 bilden die unitäre Gruppe SU (2). Außerdem ist mT = m−1 . 50. Die Abbildung m : S 3 → SU (2) mit x → m(x) ist ein Isomorphismus.
Er verstarb ein Jahr später an einem Nierenleiden.
Offensichtlich ist x = |x| „ x0 x |x| + |x| |x| |x| « = |x|(cos ϕ + ω(x) sin ϕ). 15. √Es sei x = 1 + √ 2e1 + 2e2 + e3 , dann ist |x| = 10, | x| = 3, ϕ = arccos (1/ 10) = arcsin (3/ 10). Somit erhalten wir die Darstellung √ (2e1 + 2e2 + e3 ) 3 1 √ x = 10 √ + 3 10 10 und sehen auch, dass eine Berechnung keinerlei Probleme macht. 16 (Formel von de Moivre). Es seien x = x0 + x ∈ H, x = 0, n ∈ Z; dann gilt: (cos ϕ + ω(x) sin ϕ)n = cos nϕ + ω(x) sin nϕ. Beweis. Wir bemerken zunächst, dass gegenüber dem komplexen Fall die imaginäre Einheit i hier durch ein Element der 2-dimensionalen Einheitssphäre ersetzt werden muss.