Cross-over in non-standard random-matrix spectral fluctuations without unfolding

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.