A system of real quaternion matrix equations with applications

Let HH be the real quaternion algebra and Hn×mHn×m denote the set of all n×mn×m matrices over HH. Let P∈Hn×nP∈Hn×n and Q∈Hm×mQ∈Hm×m be involutions, i.e., P2=I,Q2=IP2=I,Q2=I. A matrix A∈Hn×mA∈Hn×m is said to be (P,Q)(P,Q)-symmetric if A=PAQA=PAQ. This paper studies the system of linear real quaternion matrix equationsA1X1=C1X1B1=C2A2X2=C3X2B2=C4A3X1B3+A4X2B4=Cc.We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied. As applications, we discuss the necessary and sufficient conditions for the systemAaX=Ca,XBb=Cb,AcXBc=Ccto have a (P,Q)(P,Q)-symmetric solution. We also show an expression of the (P,Q)(P,Q)-symmetric solution to the system when the solvability conditions are met. Moreover, we provide an algorithm and a numerical example to illustrate our results. The findings of this paper extend some known results in the literature.