Recently, the singular value decomposition (SVD) was applied to standard Gaussian ensembles of Random Matrix Theory (RMT) to determine the scale invariance in the spectral fluctuations without performing any unfolding procedure. Here, SVD is applied
directly to the $ u$-Hermite ensemble and to a sparse matrix ensemble, decomposing the corresponding spectra in trend and fluctuation modes. In correspondence with known results, we obtain that fluctuation modes exhibit a cross-over between soft and rigid behavior. By using the trend modes we performed a data-adaptive unfolding, and we calculate traditional spectral fluctuation measures. Additionally, ensemble-averaged and individual-spectrum averaged statistics are calculated consistently within the same basis of normal modes.
The statistics of random-matrix spectra can be very sensitive to the unfolding procedure that separates global from local properties. In order to avoid the introduction of possible artifacts, recently it has been applied to ergodic ensembles of Rando
m Matrix Theory (RMT) the singular value decomposition (SVD) method, based on normal mode analysis, which characterizes the long-range correlations of the spectral fluctuations in a direct way without performing any unfolding. However, in the case of more general ensembles, the ergodicity property is often broken leading to ambiguities between spectrum-unfolded and ensemble-unfolded fluctuation statistics. Here, we apply SVD to a disordered random-matrix ensemble with tunable nonergodicity, as a mathematical framework to characterize the nonergodicity. We show that ensemble-averaged and individual-spectrum averaged statistics are calculated consistently using the same normal mode basis, and the nonergodicity is explained as a breakdown of this common basis.
