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Spectra of ordered eigenvalues of finite Random Matrices are interpreted as a time series. Dataadaptive techniques from signal analysis are applied to decompose the spectrum in clearly differentiated trend and fluctuation modes, avoiding possible artifacts introduced by standard unfolding techniques. The fluctuation modes are scale invariant and follow different power laws for Poisson and Gaussian ensembles, which already during the unfolding allows to distinguish the two cases.PACS numbers: 05.45. Tp,05.45.Mt,02.50.Sk The study of spectral fluctuations within the framework of Random Matrix Theory (RMT) is a standard tool in the statistical study of quantum chaos in the excitation spectra of quantum systems [1][2][3][4]. Recently, the approach has found new applications in many fields, such as in the study of eigenspectra of adjacency matrices of networks [5][6][7], and eigenspectra of empirical correlation matrices in finance [8][9][10], the climate [11], electro-and magnetoencephalography [12][13][14], and in complex systems [15]. The interest of the approach lies in the fact that the level density fluctuations ρ(E) = ρ(E) − ρ(E) around the smooth global density ρ(E) are universal and indicate the underlying symmetry class of the system [2,16]. On the other hand, the global level density ρ(E) is system dependent, and an unfolding procedure needs to be performed, to separate the global and the fluctuating parts [1]. The unfolding is straightforward if an analytical formula is known to describe the global level density ρ(E) for the system under study, such as e.g. the gaussian and semicircle distributions for Possion and GOE matrix ensembles from RMT [2], or the MarchenkoPastur distribution for the Laguerre ensemble of random Wishart correlation matrices [17]. However, such analytical formulae are formally only adequate in the asymptotic limit for spectra with an infinite number of levels. Often, an analytical form for ρ(E) is unknown, as is the case for adjacency matrices [5]. In practical cases, having finite, albeit large matrices, the usual approach is then to project the sequence of ordered eigenvalues into unfolded values E(n) → N [E(n)], using a smooth (often polynomial) approximation N (E) to the accumulated density , 5, 20]. After unfolding, the short-range and long-range correlations can be quantified using standard fluctuations measures such as the Nearest-Neighbour Spacing (NNS) distribution, number variance Σ 2 and ∆ 3 . In a recent approach, the unfolded fluctuations of the accumulated level density function N (E) = N (E) − N (E) (also called δ n function) *

We apply the singular value decomposition (SVD) method, based on normal-mode analysis, to decompose the spectra of finite random matrices of standard Gaussian ensembles in trend and fluctuation modes. We use the fact that the fluctuation modes are scale invariant and follow a power law, to characterize the transition between the extreme regular and chaotic cases. Thereby, we quantify the quantum chaos in systems described by random matrix theory in a direct way, without performing any previous unfolding procedure, and therefore, avoiding possible artifacts.

Recently, singular value decomposition (SVD) was applied to standard Gaussian ensembles of random-matrix theory to determine the scale invariance in spectral fluctuations without performing any unfolding procedure. Here, SVD is applied directly to the β-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 crossover between soft and rigid behavior. In this way, possible artifacts introduced applying unfolding techniques are avoided. By using the trend modes, we perform 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.