We investigate the spectral efficiency gain of an uplink Cognitive Radio (CR) Multi-Input-Multi-Output system in which the Secondary User (SU) is allowed to share the spectrum with the Primary User (PU) using a specific precoding scheme to communicate with a common receiver. The proposed scheme exploits, at the same time, the free eigenmodes of the primary channel after a space alignment procedure and the interference threshold tolerated by the PU. At the common receiver, we adopt a Successive Interference Cancellation (SIC) technique to eliminate the effect of the detected primary signal transmitted through the exploited eigenmodes. Furthermore, we analyze the SIC operation inaccuracy as well as the CSI estimation imperfection on the PU and SU throughputs. Numerical results show that our proposed scheme enhances considerably the cognitive achievable rate. For instance, in case of a perfect detection of the PU signal, the CR rate remains non-zero for high Signal to Noise Ratio which is usually impossible when we only employ a space alignment technique. We show that a modified water-filling power allocation policy at the PU can increase the secondary rate with a marginal degradation of the primary rate. Finally, we investigate the behavior of the PU and SU rates through the study of the rate achievable region.