In this paper, we theoretically investigate a new technique for simultaneous information and power transfer (SWIPT) in multiple-input multiple-output (MIMO) point-to-point with radio frequency energy harvesting capabilities. The proposed technique exploits the spatial decomposition of the MIMO channel and uses the eigenchannels either to convey information or to transfer energy. In order to generalize our study, we consider channel estimation error in the decomposition process and the interference between the eigenchannels. An optimization problem that minimizes the total transmitted power subject to maximum power per eigenchannel, information and energy constraints is formulated as a mixed-integer nonlinear program and solved to optimality using mixed-integer second-order cone programming. A near-optimal mixed-integer linear programming solution is also developed with robust computational performance. A polynomial complexity algorithm is further proposed for the optimal solution of the problem when no maximum power per eigenchannel constraints are imposed. In addition, a low polynomial complexity algorithm is developed for the power allocation problem with a given eigenchannel assignment, as well as a low-complexity heuristic for solving the eigenchannel assignment problem.