Two algebraic methods for least squares L-structured and generalized L-structured problems of the commutative quaternion Stein matrix equation

In this paper, we put forward two efficient methods for solving the commutative quaternion Stein matrix equation X + AXB = C. By combining real representation of a commutative quaternion matrix and H-representation or generalized H-representation of special matrices, we investigate the minimal norm least squares L-structured and generalized L-structured solutions of the previous commutative quaternion matrix equation and derive their expressions. In this way, we first present a theoretical study on extending L-structured real matrices to generalized L-structured matrices, and introduce some generalized L-structured matrices. Based on them, we then discuss their applications in commutative quaternion Stein matrix equation. The algorithms only involve real operations. Consequently, it is very simple and convenient, and it can be used for all kinds of commutative quaternion matrix equation with similar problems. Furthermore, an illustrative example is provided to show the feasibility of the given methods.