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This paper considers vector communications through multiple-input multiple-output (MIMO) channels with a set of quality of service (QoS) requirements for the simultaneously established substreams. Linear transmit-receive processing (also termed linear precoder at the transmitter and linear equalizer at the receiver) is designed to satisfy the QoS constraints with minimum transmitted power (the exact conditions under which the problem becomes unfeasible are given). Although the original problem is a complicated nonconvex problem with matrix-valued variables, with the aid of majorization theory, we reformulate it as a simple convex optimization problem with scalar variables. We then propose a practical and efficient multilevel water-filling algorithm to optimally solve the problem for the general case of different QoS requirements. The optimal transmit-receive processing is shown to diagonalize the channel matrix only after a very specific prerotation of the data symbols. For situations in which the resulting transmit power is too large, we give the precise way to relax the QoS constraints in order to reduce the required power based on a perturbation analysis. We also propose a robust design under channel estimation errors that has an important interest for practical systems. Numerical results from simulations are given to support the mathematical development of the problem.