Multi-user multiple-input multiple-output (MU-MIMO) is an important transmission technique for wireless communication systems, which increases spectral efficiency by transmitting data to several users at the same time and frequency band. The main design issue for the MU-MIMO system is to suppress co-channel interference among all users by using appropriate MIMO precoding technique. Generalized eigenvalue decomposition (GEVD) is an inevitable and high-complexity processing unit in the leakage-based precoding system with multi-antenna users. This paper presents a Givens rotation-based algorithm for hardware implementation, which can be realized by only coordinate rotation digital computer (CORDIC) processors. The proposed GEVD processor was designed for the MU-MIMO mode in the IEEE 802.11ac system. It supports eight transmitting antennas at the base station and two receiving antennas at each of four users. The GEVD processor was designed and implemented by using a TSMC 40-nm process technology. The chip synthesis results show that the proposed GEVD processor achieves a user throughput of 0.98 M matrix/sec/user, equivalent to 0.25 M channel matrices/sec, conforming to the IEEE 802.11ac standard.