Open AccessPerfect hypercomplex algebras: Semi-tensor product approach
Author(s) -
Daizhan Cheng,
AUTHOR_ID,
Zhengping Ji,
Jun-e Feng,
Shihua Fu,
Jianli Zhao,
AUTHOR_ID,
AUTHOR_ID
Publication year - 2021
Publication title -
mathematical modelling and control
Language(s) - English
Resource type - Journals
ISSN - 2767-8946
DOI - 10.3934/mmc.2021017
Subject(s) - hypercomplex number , tensor product , mathematics , invertible matrix , zero (linguistics) , commutative property , pure mathematics , product (mathematics) , associative property , algebra over a field , set (abstract data type) , zero divisor , quaternion , computer science , geometry , linguistics , philosophy , programming language
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.