A system of real quaternion matrix equations with applications

Abstract

Let H be the real quaternion algebra and H n × m denote the set of all n × m matrices over H . Let P ∈ H n × n and Q ∈ H m × m be involutions, i.e., P 2 = I , Q 2 = I . A matrix A ∈ H n × m is said to be ( P , Q ) -symmetric if A = PAQ . This paper studies the system of linear real quaternion matrix equations A 1 X 1 = C 1 X 1 B 1 = C 2 A 2 X 2 = C 3 X 2 B 2 = C 4 A 3 X 1 B 3 + A 4 X 2 B 4 = C c . 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 system A a X = C a , XB b = C b , A c XB c = C c to have a ( P , Q ) -symmetric solution. We also show an expression of the ( 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.