This paper considers a multiple-input multiple-output (MIMO) downlink communication scenario with one base station and two users, where each user is equipped with m antennas and the base station is equipped with n antennas. To efficiently exploit the spectrum resources, we propose a transmission protocol which combines generalized singular value decomposition (GSVD) and non-orthogonal multiple access (NOMA). The average data rates achieved by the two users are adopted as performance metrics for evaluation of the proposed GSVD-NOMA scheme. In particular, we first characterize the limiting distribution of the squared generalized singular values of the two users' channel matrices for the asymptotic case where the numbers of transmit and receive antennas approach infinity. Then, we calculate the normalized average individual rates of the users in the considered asymptotic regime. Furthermore, we extend the proposed GSVD-NOMA scheme to the MIMO downlink communication scenario with more than two users by using a hybrid multiple access (MA) approach, where the base station first divides the users into different groups, then the proposed GSVD-NOMA scheme is implemented within each group, and different groups are allocated with orthogonal bandwidth resources. Finally, numerical results are provided to validate the effectiveness of the proposed GSVD-NOMA protocol, and the accuracy of the developed analytical results.