Matrix and Spectral Similarity Networks
Luciano da Fontoura Costa · HAL (Le Centre pour la Communication Scientifique Directe) · 2023
Matrices are fundamentally important in science and technology thanks to their versatility for representing and modeling a wide range of natural and abstract structures and dynamics. A great deal of the properties of matrices stem from their spectral structure, namely the involved eigenvalues and eigenvectors, to the point that the systematic study of eigenvalues distribution (the matrix spectrum) has constituted a subject of great attention. In the present work, we develop an approach to visualizing and characterizing the spectral structure of square, real-valued random matrices in terms of complex networks obtained by comparing the coincidence similarity between the eigenvalues and/or eigenvectors of a given ensemble of matrices. An index to quantify how much a matrix is symmetric, corresponding to coincidence similarity index between the upper and lower triangular parts of a given matrix, is also proposed and employed to complement the study of the considered random matrices. The potential of the approach is illustrated respectively to three types of random matrix ensembles, namely gaussian (Ginibre) ensemble, uniform random matrices, and quasi-symmetric Gaussian random matrices. In all cases, the respective eigenvalues networks revealed an impressive structured heterogeneity of interrelationships between the obtained eigenvalues. Additional results of potential interest are also presented and discussed.