Characterization Of Positive Definite Matrices in Generalized Metric and Probabilistic Normed Spaces
Keywords:
Positive definite matrices, probabilistic normed spaces, generalized metrics, eigenvalue spectrum, topological indices, matrix analysis, scaling behaviourAbstract
This study explores the structural and spectral characterization of positive definite matrices defined over generalized metric spaces and probabilistic normed spaces (PMNS). By modelling connection uncertainty through probabilistic adjacency matrices, we investigate how classical topological indices reflect and predict spectral complexity. The number of active vertices V×(G), along with the Randić and harmonic indices, are shown to correlate significantly with spectral quantities like Shannon entropy, eigenvector participation, and level spacing statistics. We introduce a universal scaling parameter ξ∝n1/2that organizes the transition from sparse to dense matrix regimes across all topological and spectral measures. These findings provide new insight into the predictability and structural consistency of matrix behaviour in generalized and uncertain metric frameworks.
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