An Analysis to Some Systems of Matrix Equations with \({\phi }\)-Hermitian Solutions for Some Nonstandard Involution \({\phi }\) Over the Real Quaternion Algebra

This paper examines some systems of matrix equations with \({\phi }\)-Hermitian solutions for some nonstandard involution \({\phi }\) over the real quaternion algebra. We use the solvability conditions to determine the general solution for these systems, that is included a unique unknown. This approach entails using the Moore-Penrose inverse and the equality of coefficient matrix ranks. We employ these techniques to construct new algorithms capable of calculating general solutions. As a result, we identify the necessary and sufficient conditions for these systems’ consistency, leading to the derivation of their general solutions. We also applied these algorithms in some numerical examples to verify the theoretical outcomes.