Calculus for eight-dimensional hypercomplex algebras
Résumé
This paper introduces the basis of calculus with the formulation of partial derivatives over hypercomplex variables for two 8-dimensional hypercomplex algebras: octonions and splitbiquaternions. The work can be seen as a direct extension of Wirtinger/CR calculus for complex-numbers algebra, and HR-calculus or generalized HR-calculus for quaternion algebra. We first show that the approach based on rotations over basis components used in the development of HR-calculus or generalized HR-calculus is not suitable for the two considered 8-dimensional hypercomplex algebras. Instead, we justify the use of transformations based on Hadamard matrices, which allows a common framework for the development of partial derivatives for complex-numbers, quaternion, octonion and splitbiquaternion algebras. The results of this paper are an important step towards the development of optimization algorithms and adaptive filtering algorithms of hypercomplex variables for 8-dimensional hypercomplex algebras.
Domaines
Mathématiques [math]Origine | Fichiers produits par l'(les) auteur(s) |
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