Perfect hypercomplex algebras: Semi-tensor product approach

Daizhan Cheng; Zhengping Ji; Jun-e Feng; Shihua Fu; Jianli Zhao; Daizhan Cheng; Zhengping Ji; Jun-e Feng; Shihua Fu; Jianli Zhao

doi:10.3934/mmc.2021017

Mathematical Modelling and Control

2021, Volume 1, Issue 4: 177-187. doi: 10.3934/mmc.2021017

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Research article

Perfect hypercomplex algebras: Semi-tensor product approach

1.
Key Laboratory of Systems and Control, AMSS, Chinese Academy of Sciences, Beijing, China
2.
School of Mathematics, Shandong University, Jinan, China
3.
Research Center of Semi-tensor Product of Matrices: Theory and Appllications, Liaocheng University, Liaocheng, China

Received: 14 October 2021 Accepted: 24 December 2021 Published: 29 December 2021

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $ 2 $-dimensional PHAs are investigated. Second, all the $ 3 $-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, $ 4 $- and higher dimensional PHAs are also considered.
- perfect hypercomplex algebra (PHA),
- zero-set,
- semi-tensor product (STP) of matrices
Citation: Daizhan Cheng, Zhengping Ji, Jun-e Feng, Shihua Fu, Jianli Zhao. Perfect hypercomplex algebras: Semi-tensor product approach[J]. Mathematical Modelling and Control, 2021, 1(4): 177-187. doi: 10.3934/mmc.2021017

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Abstract

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $ 2 $-dimensional PHAs are investigated. Second, all the $ 3 $-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, $ 4 $- and higher dimensional PHAs are also considered.

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