Complex number: Difference between revisions

Revision as of 04:42, 20 September 2020

Revision as of 04:48, 20 September 2020

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The Cayley–Dickson construction is closely related to the [[regular representation]] of '''C''', thought of as an '''R'''-[[Algebra (ring theory)|algebra]] (an '''R'''-vector space with a multiplication), with respect to the basis {{math|(1, ''i'')}}. This means the following: the '''R'''-linear map

:<math>\mathbb{C} \rightarrow \mathbb{C},\quad z \mapsto wz</math>

for some fixed complex number <math>{{mvar|w \ =\ \Re(w) + i\Im(w)</math>}} can be represented by a {{math|2 × 2}} matrix (once a basis has been chosen). With respect to the basis {{math|(1, ''i'')}}, this matrix is

:<math>

\begin{pmatrix}\Re(w) & -\Im(w) \\

\Imoperatorname{Re}(w) & -\Reoperatorname{Im}(w) \\

\operatorname{Im}(w) & \;\; \operatorname{Re}(w)

\end{pmatrix}

</math>,