This paper studies amplify-and-forward (AF) multiple-input multiple-output (MIMO) beamforming relay networks based on the minimum mean square error (MMSE) criterion under various transmit power constraints. The primary contribution of this paper is the derivation of a set of optimal relay amplifying matrices and source-destination beamforming vectors under diverse conditions of transmit power constraints on the source and the relay for the AF MIMO wireless relay network. By comparing the bit error rate (BER) performance of each case,
an efficient design of a half-duplex AF relay system is presented.