The solutions of the quaternion matrix equation AX^ε + BX^δ = 0

In this paper, we discuss the quaternion matrix equation AX^ε+BX^δ=0, where ε∈{I,C}, δ∈{†,⁎} and I, C, †, ⁎ denote the identity, involutive automorphism, involutive anti-automorphism and transpose, involutive automorphism and anti-automorphism and transpose, respectively. Firstly, we transform the given matrix equation into the matrix equation A˜Y^ε+B˜Y^δ=0 with complex coefficient matrices and quaternion target matrix Y by utilizing the regularity of (A,B), where A˜=PAQ and B˜=PBQ are two complex matrices with P,Q being two invertible quaternion matrices. Secondly, we show that the solution can be obtained in terms of Q, the Kronecker canonical form of (A˜,B˜) and one of the two invertible quaternion matrices which transform (A˜,B˜) into its Kronecker canonical form. Meanwhile, we also decouple the transformed equation into some systems of small-scale equations in terms of Kronecker canonical form of (A˜,B˜). Thirdly, we determine the dimension of the solution space of the equation in terms of the sizes of the blocks arising in the Kronecker canonical form. Finally, we give the necessary and sufficient condition for the existence of the unique solution.