Least-Norm of the General Solution to Some System of Quaternion Matrix Equations and Its Determinantal Representations

Abdur Rehman,Ilyas Ali,Muhammad Akram,Abdul Shakoor,Ivan Kyrchei

doi:10.1155/2019/9072690

Abstract

We constitute some necessary and sufficient conditions for the system A1X1=C1, X1B1=C2, A2X2=C3, X2B2=C4, A3X1B3+A4X2B4=Cc, to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

Highlights

The notation R is reserved for the real number field and H푚×푛 stands for the set of all m × n matrices over the quaternion skew field
Determinantal representation of a solution gives a direct method of its finding analogous to the classical Cramer’s rule that has important theoretical and practical significance
Song et al [58] have just recently considered determinantal representations of general solution to the two-sided coupled generalized Sylvester matrix equation over H obtained using the theory of row-column determinants as well

Summary

Introduction

The notation R is reserved for the real number field and H푚×푛 stands for the set of all m × n matrices over the quaternion skew field. Determinantal representation of a solution gives a direct method of its finding analogous to the classical Cramer’s rule that has important theoretical and practical significance Through looking for their more applicable explicit expressions, there are various determinantal representations of generalized inverses even with the complex or real entries, in particular for the Moore-Penrose inverse (see, e.g., [42,43,44]). Song et al [58] have just recently considered determinantal representations of general solution to the two-sided coupled generalized Sylvester matrix equation over H obtained using the theory of row-column determinants as well. Their proposed approach differs from our proposed.

Preliminaries

Conclusion