This letter proposes a multi-input multi-output (MIMO) transceiver design termed channel bi-diagonalization (CBD), whose precoder and equalizer bi-diagonalize the channel matrix. By representing the bidiagonal MIMO channel with a trellis diagram, we can apply the Viterbi algorithm for the maximum likelihood (ML) detection and decoding, or the BCJR algorithm for the maximum a posteriori (MAP) detection and decoding, which drastically simplifies the optimal MIMO communications. Simulation results verify the superior performance of the proposed CBD scheme.