Some Properties of Reduced Biquaternion Tensors

Abstract

Compared to quaternions, reduced biquaternions satisfy the multiplication commutative rule and are widely employed in applications such as image processing, fuzzy recognition, image compression, and digital signal processing. However, there is little information available regarding reduced biquaternion tensors; thus, in this study, we investigate some properties of reduced biquaternion tensors. Firstly, we introduce the concept of reduced biquaternion tensors, propose the real and complex representations of reduced biquaternion tensors, and prove several fundamental theorems. Subsequently, we provide the definitions for the eigenvalues and eigentensors of reduced biquaternion tensors and present the Gersˇgorin theorem as it applies to their eigenvalues. Additionally, we establish the relationship between the reduced biquaternion tensor and its complex representation. Notably, the complex representation is a symmetry tensor, which significantly simplifies the process and complexity of solving for eigenvalues. Corresponding numerical examples are also provided in the paper. Furthermore, some special properties of eigenvalues of reduced biquaternion tensors are presented.

Keywords

• reduced biquaternion tensors
• eigenvalues and eigentensors
• complex representation