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

Abstract

We constitute some necessary and sufficient conditions for the system $A_{\mathrm{1}}{X}_{\mathrm{1}}=C_{\mathrm{1}}$, $X_{\mathrm{1}}{B}_{\mathrm{1}}=C_{\mathrm{2}}$, $A_{\mathrm{2}}{X}_{\mathrm{2}}=C_{\mathrm{3}}$, $X_{\mathrm{2}}{B}_{\mathrm{2}}=C_{\mathrm{4}}$, $A_{\mathrm{3}}{X}_{\mathrm{1}}{B}_{\mathrm{3}}+A_{\mathrm{4}}{X}_{\mathrm{2}}{B}_{\mathrm{4}}=C_c$, 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.