A Pair of Hermitian and Anti-Hermitian Solutions of a Generalized Sylvester Matrix Equation over Commutative Quaternion Algebra

Key Points

• The aim is to explore solutions to a generalized Sylvester matrix equation in quaternion algebra.
• Investigated solutions using a complex representation of quaternion matrices.
• Applied the Kronecker product and vec-operation for matrix manipulation.
• Utilized Moore–Penrose generalized inverse for solution derivation.
• Established necessary and sufficient conditions for solution existence.
• Derived explicit expressions when solutions are solvable.
• Identified criteria for the existence of Hermitian and anti-Hermitian solutions.
• Derived explicit solution forms for solvable conditions.
• Provided numerical examples illustrating the findings.

Abstract