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Pré-Publication, Document De Travail Année : 2022

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.
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Dates et versions

hal-03558753 , version 1 (04-02-2022)
hal-03558753 , version 2 (02-12-2022)
hal-03558753 , version 3 (20-02-2023)

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  • HAL Id : hal-03558753 , version 2

Citer

Martin Bouchard, Juhi Khalid. Calculus for eight-dimensional hypercomplex algebras. 2022. ⟨hal-03558753v2⟩
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