Autonomous non-linear systems, often arising in different fields of science, are usually difficult to analyse. Pseudo-linear (PL) representation of such a non-linear system has become very popular recently to deal with this difficulty, by applying the well-established linear system theory tools to non-linear systems. This paper presents a comprehensive analysis of non-linear autonomous systems using eigenstructure-based analysis through the PL form representation. The sophisticated ability of non-linear eigenvalues for qualitative behaviour determination of the non-linear systems is fully discussed. Knowing the fact that the PL form of a given non-linear system is not unique, this paper presents a systematic method for obtaining infinitely many PL forms for a given non-linear system by introducing a new concept of basis set for the space of PL forms. Looking to the counterexamples published in the literature, a conclusive proposition is established, which apparently provides a framework to attack the long struggle in the stability consideration of non-linear systems via a PL form by applying the crucial role of non-linear eigenvectors neglected in the previous studies. Some illustrative examples are also presented to better highlight the merit of the proposed method. The simulations are carried out by MATLAB 2009b.