Multi-input multi-output (MIMO) cognitive radio (CR) network is an emerging technology which is expected to be a potential solution to enhance the utilization efficiency of the radio spectrum. CR network supports opportunistic spectrum sharing by allowing the secondary users to share the same radio spectrum occupied by the primary users. However, the spectrum sharing by the secondary users should be performed only when the Quality-of-Service (QoS) degradations of primary users due to the spectrum sharing are carefully controlled. The traditional design of the secondary user transmit covariance is to maximize its mutual information while satisfying the interference power constraints induced at the primary receivers. However, in many practical applications, we would like to limit the number of independent data streams of the transmitter to take advantage of the transmit diversity or due to the restrictions from the top layers in the network. In this paper, we focus on a rank-constrained secondary user transmit covariance design problem. We first derive a structural result on the optimal solution of the rank-constrained problem. Based on the solution structure, we transform the rank-constrained problem into an equivalent problem with a unitary constraint. After that, we derive an iterative projected steepest descent algorithm which can converge to a local optimal solution. Furthermore, we shall show that under some special cases, we could even derive closed-form global optimal solution. The numerical results show the superior performance of the our proposed technique over the baseline schemes, the rank relaxation based randomization technique.