| Summary: | We wrote this paper to study in detail the mathematical methods used in deriving the parameters p'_1, p'_2, x'_1, x2_2, M, K and exp(2n) in the research papers by Han et al. (1999), Makarov (2018a), Han et al. (1993), and Han et al. (1995) to find the Schmidt modes A_K in quantum entanglement. Here, we have analysed and developed a thorough calculation in exploring the rationales behind the existence of these parameters in two and three-coupled harmonic oscillators. Various mathematical approaches were applied in the study, ranging from polynomials and linear algebra to trigonometry and the Pythagorean theorem. We found the parameters K and exp(2n) using the matrices’ determinants and eigenvalues. With these rationales in deriving the parameters K and exp(2n) in the research papers for the two-coupled harmonic oscillators, we have formulated similar parameters for three-coupled harmonic oscillators as the conclusion of our study.
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