  Our research highlights the effectiveness of utilizing matrices akin to
Wishart matrices, derived from magnetization time series data under specific
dynamics, to elucidate phase transitions and critical phenomena in the Q-state
Potts model. By employing appropriate statistical methods, we not only discern
second-order transitions but also differentiate weaker first-order transitions
through careful analysis of the density of eigenvalues and their fluctuations.
Furthermore, we investigate the method's sensitivity to stronger first-order
transition points. Importantly, we establish a robust correlation between the
system's actual thermodynamics and the spectral thermodynamics encapsulated
within the eigenvalues. Our findings are further substantiated by correlation
histograms of the time series data, revealing insightful patterns. Expanding
upon our core findings, we present a didactic analysis that draws parallels
between the spectral properties of criticality in a spin system and matrices
intentionally imbued with correlations (a toy model). Within this framework, we
observe a universal behavior characterized by the distribution of eigenvalues
into two distinct groups, separated by a gap dependent on the level of
correlation, influenced by temperature-induced changes in the spin system.
