Eigenvalue Decomposition and Its Application to Markov Chain Analysis

Mathematics — 2025, Undergraduate

This study examined eigenvalue decomposition and applied it to the analysis of Markov chains. Transition matrices were decomposed, steady-state vectors were derived from dominant eigenspaces, and convergence rates were interpreted. The findings revealed that eigenvalue analysis determined long-run behaviour of the chains accurately, with the dominant eigenvalue governing stability. The study observed that spectral properties explained mixing behaviour. It recommends teaching eigenvalue methods as a bridge between abstract algebra and applied stochastic modelling.

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