On Perfect Hypercomplex Algebra

TL;DR

This paper investigates the structure and properties of perfect hypercomplex algebras (PHAs), providing conditions for their characterization, analyzing zero sets across dimensions, and exploring matrices over PHAs.

Contribution

It offers a comprehensive review of conditions for PHAs via semi-tensor products and explores their zero sets and matrices in various dimensions, extending understanding of hypercomplex algebra.

Findings

Characterization of PHAs using semi-tensor product

Explicit zero sets for 2D and 3D PHAs

Properties of matrices over PHAs

Abstract

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebra (PHA) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product(STP) of matrices are reviewed. The zero set is defined for non-invertible hypercomplex numbers in a given PHA, and a characteristic function is proposed for calculating zero set. Then PHA of different dimensions are considered. First, $2$-dimensional PHAs are considered as examples to calculate their zero sets etc. Second, all the $3$-dimensional PHAs are obtained and the corresponding zero sets are investigated. Third, $4$-dimensional or even higher dimensional PHAs are also considered. Finally, matrices over pre-assigned PHA, called perfect hypercomplex matrices (PHMs) are considered. Their properties are also investigated.