This paper considers a cognitive radio (CR) network, in which the unlicensed (secondary) users (SUs) are allowed to concurrently access the spectrum allocated to the licensed (primary) users, provided that the interference of SUs with the primary users (PUs) satisfies certain constraints. It is more general and owns a stronger challenge to ensure the quality of service (QoS) of PUs, as well as to maximize the sum-rate of SUs. On the other hand, the multiple-antenna mobile user case has not been well investigated for the target problem in the open literature. We refer to this setting as multiple input multiple output multiple access channels (MIMO-MAC) in the CR network. Subject to the interference constraints of SUs and the peak power constraints of SUs, the sum-rate maximization problem is solved. To efficiently maximize the achievable sum-rate of SUs, a tight pair of upper and lower bounds, as an interval, of the optimal Lagrange multiplier is proposed. It can avoid ineffectiveness or inefficiency when the dual decomposition is used. Furthermore, a novel water-filling-like algorithm is proposed for the inner loop computation of the proposed problem. It is shown that this algorithm used in the inner loop computation can obtain the exact solution with a few finite computations, to avoid one more loop, which would be embedded in the inner loop. In addition, the proposed approach overcomes the limitation of Hermitian matrices, as optimization variables. This limitation to the optimization problem in several complex variables has not been well investigated so far. As a result, our analysis and results are solidly extended to the field of complex numbers, which are more compatible with practical communication systems.