Analysis of Solvability and Representation of General Solutions for Anti-Hermitian Constrained Quaternion Matrix Equations

Given the extensive application of anti-Hermitian matrices in engineering and the sciences, this paper presents the general solution to a constrained system of matrix equations that incorporate their anti-Hermitian property. The primary focus of this study is to solve this constrained system of quaternion matrix equations. Solvability conditions are established using rank equalities, and explicit representation formulas are provided, employing the Moore-Penrose inverse of coefficient matrices and their projections. An algorithm and a numerical example are presented to validate the research findings. The numerical example utilizes a unique direct method for obtaining solutions to the given system, based on exclusive determinantal representations of the Moore-Penrose inverse within the noncommutative row-column determinant theory recently developed by one co-author. The results obtained demonstrate significant novelty, even when applied to the corresponding complex matrix equations as a special case.