Special solutions to a matrix equation over the split quaternions

Abstract

Split quaternions, as an extension of classical quaternions, exhibit distinct algebraic properties and offer valuable applications across various fields. This paper investigates solutions to the split quaternion matrix equation $\sum_{i=1}^{l}A_{i}X_{i}B_{i}= C$, providing necessary and sufficient conditions for its solvability and expressions for various solution types. Solvability criteria and explicit solution forms are derived for general, pure imaginary, and real solutions. Additionally, corresponding conditions and expressions are presented for (skew-)centrohermitian solutions. To illustrate the theoretical results, numerical examples and algorithms are provided.