Multi-dimensional chaotic systems were reduced to low-dimensional space embedded equations (SEEs), and their macroscopic and statistical properties were investigated using eigen analysis of the moment vector equation (MVE) of the SEE. First, the state space of the target system was discretized into a finite discrete space. Next, an embedding from the discrete space to a low-dimensional discrete space was defined. The SEE of the target system was derived using the embedding. Finally, eigen analysis was applied to the MVE of the SEE to derive the properties of the target system. The geometric increase in the dimension of the MVE with the dimension of the target system was avoided by using the SEE. The pdfs of arbitrary elements in the target nonlinear system were derived without a reduction in accuracy due to dimension reduction. Moreover, since the dynamics of the system were expressed by the eigenvalues of the MVE, it was possible to identify multiple steady states that cannot be done using numerical simulation. This approach can thus be used to analyze the macroscopic and statistical properties of multi-dimensional chaotic systems.