There have been many results on designing transceivers for MIMO channels. In early results, the transceiver is designed for a given bit allocation. In this paper we will jointly design the transceiver and bit allocation for maximizing bit rate. By using a high bit rate assumption, we will see that the optimal transceiver and bit allocation can be obtained in a closed form using simple Hadamard inequality and the Poincaré separation theorem. In the simulation, we will demonstrate the usefulness of the joint design. Simulation results, in which a high bit rate assumption is not used in allocating bits, show that a higher bit rate can be achieved compared to previously reported methods.