A System of Tensor Equations over the Dual Split Quaternion Algebra with an Application

Abstract

In this paper, we propose a definition of block tensors and the real representation of tensors. Equipped with the simplification method, i.e., the real representation along with the M-P inverse, we demonstrate the conditions that are necessary and sufficient for the system of dual split quaternion tensor equations (A∗NX,X∗SC)=(B,D), when its solution exists. Furthermore, the general expression of the solution is also provided when the solution of the system exists, and we use a numerical example to validate it in the last section. To the best of our knowledge, this is the first time that the aforementioned tensor system has been examined on dual split quaternion algebra. Additionally, we provide its equivalent conditions when its Hermitian solution X=X∗ and η-Hermitian solutions X=Xη∗ exist. Subsequently, we discuss two special dual split quaternion tensor equations. Last but not least, we propose an application for encrypting and decrypting two color videos, and we validate this algorithm through a specific example.

Keywords

• dual split quaternion tensor equation
• Einstein product
• real representation
• η-Hermitian solution
• Moore–Penrose inverse