Analysis of solvability and representation of general solutions for anti-Hermitian constrained quaternion matrix equations

This paper addresses the constrained system of quaternion matrix equations incorporating anti-Hermitian properties, driven by the significance of symmetric solutions in diverse applications. Solvability conditions are determined via rank equalities and relationships derived from Moore–Penrose inverses and induced projectors. Explicit solution representations are obtained, utilizing the Moore–Penrose inverse and projections. The originality of the results is established through a novel technique based on quaternion row-column determinant theory, supported by a numerical validation. This approach retains its innovative character even when extended to complex matrix equations using conventional determinants.