The conventional method of designing the desired beampattern in MIMO systems involves the computation of the waveform covariance and/or weight matrices. In this work, through some simple derivations, it is demonstrated that there are infinitely many covariance matrices that yield the same beampattern. Furthermore, when beam is created using weighted sums of orthogonal waveforms, there are infinitely many weight matrices that also generate the same beampattern. Thus, the conditions for the beampattern invariance are formulated with respect to the covariance and the weight matrices. This theoretical foundation allows the transmitted waveform to be altered, without changing the beampattern and the orthogonal radar waveforms. Methods for the computation of beampattern-invariant covariance and weight matrices are proposed. Consequently, it is demonstrated that there is additional degrees of freedom in the design space of many applications, such as the dual-function radar communication (DFRC), in which information can be embedded in radar transmissions while keeping the radar function intact. Other potential applications are peak-to-average power ratio (PAPR) reduction at the radar transmitter and deceptive jamming avoidance.