Plasmonic optical circuits hold great promise in reducing device footprints by orders of magnitude and enabling high device complexity. The decomposition of an arbitrary linear transformation using unitary beam splitters is well-known. However, because of the inherent lossy nature of plasmonic devices, this decomposition is not useful for the practical design of devices. In this Letter, we provide a method to design an arbitrary unitary transformation using plasmonic beam splitters, which takes into account the inherent lossy nature of plasmonic modes in the decomposition process itself, while preserving the fidelity of the transformation. We do this by selecting the loss in each arm of the beam splitters and the interconnects. We also show how this method can be extended for the case of any linear transformation by extending the singular value decomposition. This method is applicable to plasmonic and waveguide-based lossy beam splitters.