What question did this study set out to answer?

The aim is to explore solutions to a generalized Sylvester matrix equation in quaternion algebra.

April 10, 2026Open Access

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

In this paper, we investigate a pair of Hermitian and anti-Hermitian solutions of a generalized Sylvester matrix equation AXB+CX¯D+EYF=G over commutative quaternion algebra by using a complex representation of commutative quaternion matrices, the Kronecker product, vec-operation, and Moore–Penrose-generalized inverse. We establish the necessary and sufficient conditions for the existence of solutions. Moreover, we derive explicit expressions when they are solvable. We also provide two numerical examples to illustrate the main results.

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