Analysis of the Principal Components of Correlation Matrices of S&P 500 Financial Data from an Econophysics Perspective

This thesis analyzes the collective dynamics of the S&P 500 from an econophysics perspective, treating the financial market as a complex system. Using daily logarithmic returns of 430 companies, time-dependent Pearson correlation matrices are constructed over sliding windows and their spectral structure is studied through the leading eigenvalues and eigenvectors. Market States (MS) are identified by applying k-Means clustering to several spectral quantities: the largest eigenvalue (interpreted as the State of the Market), the squared entries of the dominant eigenvector (read as a relative participation of each stock), and truncated matrix reconstructions C^l built from the l largest eigenpairs and normalized to retain a correlation-matrix interpretation. The stability of the resulting clusters is assessed, transition matrices and their stationary vectors are computed, and the inverse participation ratio is used to quantify how participation spreads across assets. The COVID-19 episode emerges as an atypical state in the full correlation matrices; reconstructing it requires more than one principal component, and the C^2 and C^3 constructions reproduce it, particularly for window length q=40 and k=5. During crisis periods the relative participation distributes over a larger number of assets, while the 2017-2018 interval contributes to the dynamics without appearing as an isolated regime. The analysis is descriptive rather than predictive, and provides conceptual and methodological tools relevant to portfolio design and trading strategies, with natural extensions to other international exchanges.