In this paper, a new hybrid precoding and combining method is proposed for the downlink of multiuser multiple-input multiple-output (MU-MIMO) millimeter wave (mmWave) channels. The proposed method designs the precoders and combiners for radio frequency (RF) and baseband processing, respectively, based on the minimum mean square error (MMSE) criterion and the rate fairness among users. To design the RF precoder and combiners implemented by phase shifters, a new matrix factorization algorithm is devised by combining the gradient method with the orthogonal projection. Under the total transmit power constraint, the proposed factorization method increases the achievable rate by making the columns of the RF precoder near-orthogonal and growing the Frobenius norm of the baseband precoder. In addition, a new MMSE-based rate balancing algorithm is proposed to design the baseband precoder and combiners in terms of maximizing the minimum user rate. The proposed rate balancing scheme iteratively updates the baseband precoder, the transmit power constraint for the baseband precoder, the baseband combiners, and the weighting vector for rate balancing. Through theoretical analysis, it is shown that the proposed design method has a polynomial complexity order. Numerical simulations present that the proposed matrix factorization method outperforms existing schemes requiring low computational complexity and the proposed rate balancing scheme converges to a stationary point satisfying the total transmit power constraint. Moreover, simulation results in MU-MIMO channels are provided to show that the proposed design scheme performs better than existing hybrid processing schemes while achieving the minimum user rate close to the upper bound of MMSE processing.