This paper considers a multiple-input multiple-output non-orthogonal multiple access (NOMA) downlink transmission system with different linear beamforming techniques, where the base station uses each cluster to serve a pair of users. In the considered NOMA cluster, we first derive the performance analysis of the system that uses a proposed user paring method, which exploits the different large-scale channel qualities of users to allocate the transmit power of the strong and weak users in each pair, to ensure that both users in each pair can contribute the best on the system performance. We further formulate a sum spectral efficiency (SE) maximization with a subject to the limited transmit power budget, which is emphasized to be non-convex. We have, then, proposed a framework that solves the above non-convex problem into two steps: lower bound this non-convex problem by a geometric program by using the arithmetic mean-geometric mean inequality and, then, employ the successive optimization approach to find the local Karush-Kuhn-Tucker point. Numerical results manifest that a NOMA-based network with zero forcing (ZF) beamforming gives the highest sum SE, while regularized ZF brings benefits to the SE of weak users.