Advancing Computational Tools for Analyzing Commutative Hypercomplex Algebras

TL;DR

This paper introduces a new family of commutative hypercomplex algebras called (alpha,beta)-tessarines, along with theoretical and computational tools for matrix operations, spectral analysis, and applications like image reconstruction.

Contribution

It develops the first comprehensive set of matrix operations and spectral theory for (alpha,beta)-tessarines, extending hypercomplex algebra analysis.

Findings

New algebraic system: (alpha,beta)-tessarines.

Effective matrix computation methods within this algebra.

Successful application to image reconstruction and face recognition.

Abstract

Commutative hypercomplex algebras offer significant advantages over traditional quaternions due to their compatibility with linear algebra techniques and efficient computational implementation, which is crucial for broad applicability. This paper explores a novel family of commutative hypercomplex algebras, referred to as (alpha,beta)-tessarines, which extend the system of generalized Segre's quaternions and, consequently, elliptic quaternions. The main contribution of this work is the development of theoretical and computational tools for matrices within this algebraic system, including inversion, square root computation, LU factorization with partial pivoting, and determinant calculation. Additionally, a spectral theory for (alpha,beta)-tessarines is established, covering eigenvalue and eigenvector analysis, the power method, singular value decomposition, rank-k approximation, and the pseudoinverse. Solutions to the classical least squares problem are also presented. These results not only enhance the fundamental understanding of hypercomplex algebras but also provide researchers with novel matrix operations that have not been extensively explored in previous studies. The theoretical findings are supported by real-world examples, including image reconstruction and color face recognition, which demonstrate the potential of the proposed techniques.