The theory governing breaking the reciprocity in acoustic metamaterials by using active eigen-structure control strategy is presented. Such theoretical foundation aims at demonstrating the ability of introducing controlled attenuation (or amplification) of the flow energy of acoustic waves along one particular propagation direction, in an acoustic metamaterial, while generating an amplification (or attenuation) when the propagation direction is reversed. This non-reciprocal transmission of the acoustic energy can be achieved in a flexible manner by just programming the metamaterial rather than by the alteration of the hard wiring of the components of the metamaterial. The developed theory is based on scaling and shaping the eigenvectors of the closed-loop system, relative to the open-loop system, to achieve any desirable attenuation or amplification patterns between various locations in the metamaterial during forward and backward propagations. Closed-form expressions are derived, using the linear control theory, for the transfer functions governing the transmission of waves between sources and receivers during forward and backward transmissions as a function of the eigenvector scaling parameters. These transfer functions clearly demonstrate the ability to break the reciprocity when the eigen-structure controller is used. Numerical examples are presented to demonstrate the merits and capabilities of the proposed approach in controlling the spatial distribution of the acoustic energy in one-dimensional acoustic ducts. During this entire process, the system remains behaving in a linear fashion. Generalization of the presented strategies to two-dimensional acoustic systems is a natural extension of the present work.