This book addresses both the basic and applied aspects of the finite iterative algorithm, CGLS iterative algorithm, and explicit algorithm to some linear matrix equations. The author presents the latest results in three parts: (1) We consider the finite iterative algorithm to the coupled transpose matrix equations and the coupled operator matrix equations with sub-matrix constrained. These two finite iterative algorithms are closely related and progressive. (2) We present MCGLS iterative algorithm for studying least squares problems to the generalized Sylvester-conjugate matrix equation, the generalized Sylvester-conjugate transpose matrix equation, and the coupled linear operator systems, respectively. (3) Compared with the previous two parts, we consider here the explicit solution to some linear matrix equations, which are the nonhomogeneous Yakubovich matrix equation, the nonhomogeneous Yakubovich transpose matrix equation, and the generalized Sylvester matrix equation, respectively. This book is intended for students, researchers, and professionals in the field of numerical algebra, linear matrix equations, nonlinear matrix equations, and control theory.