In this paper we consider distributed adaptive stabilization for uncertain multivariable linear systems with a time-varying diagonal matrix gain. We show that uncertain multivariable linear systems are stabilizable by diagonal matrix high gains if the system matrix is an H-matrix with positive diagonal entries. Based on matrix measure and stability theory for diagonally dominant systems, we consider two classes of uncertain linear systems, and derive a threshold condition to ensure their exponential stability by a monotonically increasing diagonal gain matrix. When each individual gain function in the matrix gain is updated by state-dependent functions using only local state information, the boundedness and convergence of both system states and adaptive matrix gains are guaranteed. We apply the adaptive distributed stabilization approach to adaptive synchronization control for large-scale complex networks consisting of nonlinear node dynamics and time-varying coupling weights. A unified framework for adaptive synchronization is proposed that includes several general design approaches for adaptive coupling weights to guarantee network synchronization.