Quaternion Two-Sided Matrix Equations with Specific Constraints

Abstract

This chapter is devoted to the survey of quaternion restricted two-sided matrix equation AXB = D and approximation problems related with it. Unique solutions to the considered approximation matrix problems and the restricted quaternion two-sided matrix equations with specific constraints are expressed in terms of the core-EP inverse and the dual core-EP inverse, the MPCEP and ∗CEPMP inverses, and the DMP and MPD inverses. The MPCEP-∗CEPMP inverses and the DMP-MPD inverses are generalized inverses obtained by combining the Moore-Penrose (MP-)inverse with the core-EP (CEP-)inverse and the MP-inverse with the Drazin (D-)inverse, respectively. Several particular cases of these equations and approximation matrix problems are presented too. Cramer’s rules for solving these constrained quaternion matrix equations and approximation matrix problems with their particular cases are developed by using of noncommutative row-column determinants. As a consequence, Cramer’s rules for solving these constrained matrix equations with complex matrices are derived as well. Numerical examples are given to illustrate gained results.