A matrix formalism defined in the complete dynamic phase space is developed to analyze spin relaxation in the East model and the dynamic slow-down of dissipative systems in general. The truncated basis set expansion provides a direct route to calculate spin correlation functions systematically and to evaluate the mean relaxation time. Examining the relaxation time scales of linear and nonlinear modes leads to the observation that the full correlation and irreducible correlation functions defined in the complete space describe the slow dynamics of a dissipative system and can be related to their equivalent physical quantities in nondissipative systems, whereas the reduced correlation functions and associated memory kernels defined in the projected space involve faster time scales and cannot be directly reduced through mode coupling approximations. Matrix relations allow us to recover the simple mode coupling and extended mode coupling equations first obtained through an elegant diagrammatic expansion by Pitts and Anderson (J. Chem. Phys. 2001, 114, 110 1). These mode coupling approaches are extended to low temperatures by analyzing higher order nonlinear modes and correcting mode coupling closures with the asymptotic behavior. Further, the second-order full correlation function can be clearly separated into the short time regime evaluated by basis set expansion and the longtime regime described by a stretched exponential arising from domain dynamics, and the resulting single-spin self-correlation function agrees with simulations over the whole temporal range.