  Random matrix theory, particularly using matrices akin to the Wishart
ensemble, has proven successful in elucidating the thermodynamic
characteristics of critical behavior in spin systems across varying interaction
ranges. This paper explores the applicability of such methods in investigating
critical phenomena and the crossover to tricritical points within the
Blume-Capel model. Through an analysis of eigenvalue mean, dispersion, and
extrema statistics, we demonstrate the efficacy of these spectral techniques in
characterizing critical points in both two and three dimensions. Crucially, we
propose a significant modification to this spectral approach, which emerges as
a versatile tool for studying critical phenomena. Unlike traditional methods
that eschew diagonalization, our method excels in handling short timescales and
small system sizes, widening the scope of inquiry into critical behavior.
