We present a comparative study of spectral complexity in Hermitian and non-Hermitian random matrix ensembles using a suite of information-theoretic quantifiers. Central to our analysis is the rigidity ratio \(R[P] = I[P]/S[P]\) , defined as the Fisher information divided by the Shannon entropy of a spectral probability distribution P(s). This dimensionless indicator captures the balance between local sharpness and global disorder and provides a scalar measure of spectral stiffness. Applying this framework to the Brody distribution, which interpolates between Poisson (integrable) and Wigner–Dyson (chaotic) statistics, we identify a non-monotonic peak in R[P] near \(\beta \approx 0.45\) . This peak reveals a regime of maximal spectral complexity—where spectral structure is most information-rich—that is not detected by conventional metrics. We extend our analysis to the Ginibre ensemble, relevant for non-Hermitian systems, and find that it exhibits even higher rigidity, geometric curvature, and disequilibrium than its Hermitian counterparts, reflecting enhanced spectral correlations in the complex plane. Our results establish R[P] as a versatile probe of complexity and transition structure in spectral statistics, unifying local and global features within a compact information-theoretic formalism. The framework developed here offers new diagnostic tools for applications ranging from quantum chaos and thermalization to decoherence and spectral analysis in open quantum systems.
