Non-water suppression MRS (NWS MRS) has several advantages. First, the unsuppressed water signal can be used as internal calibration for metabolite quantification and as a reliable frequency/phase reference for retrospective motion correction. Second, it avoids the potential artifacts caused by incomplete water suppression (WS) and extra radiofrequency deposition from WS pulses. However, the frequency modulation (FM) sidebands originating from a large water signal will distort the spectrum. Among the methods proposed to solve the problems caused by FM sidebands, post-acquisition processing methods are superior in flexibility for general use compared with experimental methods. In this study, we propose two algorithms based on advanced matrix decomposition to remove the FM sidebands. These methods, the simultaneous diagonalization (QZ) algorithm and its subsequent variant, the simultaneously generalized Schur decomposition (SGSD) algorithm, were numerically evaluated using computer simulations. In addition, we quantitatively compared the performance of these methods and the modulus method in an in vitro experiment and in vivo NWS MRS against conventional WS data. Our results show that the proposed SGSD algorithm can reduce the FM sidebands to achieve superior estimation of concentration on three major metabolites. This method can be applied directly to spectra pre-acquired under various experimental conditions without modifying the acquisition sequences.