filtering and frequency interpretations of singular spectrum analysis new filtering and spectral interpretations of singular spectrum analysis ssa are provided it is shown that the variables reconstructed from diagonal averaging of reduced rank approximations to the trajectory matrix can be obtained from a noncausal convolution filter with zero phase characteristics the reconstructed variables are readily constructed using a two pass filtering algorithm that is well known in the signal processing literature when the number of rows in the trajectory matrix is much larger than number of columns many results reported in the signal processing literature can be used to derive the properties of the resulting filters and their spectra new features of the reconstructed series are revealed using these results two examples are used to illustrate the results derived in this paper

gevd based low rank approximation for distributed adaptive node specific signal estimation in wireless sensor networks in this paper we address the problem of distributed adaptive estimation of node specific signals for signal enhancement or noise reduction in wireless sensor networks with multi sensor nodes the estimation is performed by a multi channel wiener filter mwf in which a low rank approximation based on a generalized eigenvalue decomposition gevd is incorporated in non stationary or low snr conditions this gevd based mwf has been demonstrated to be more robust than the original mwf in a centralized realization where a fusion center has access to all the nodes sensor signal observations the network wide sensor signal correlation matrices and the low rank approximation can be directly estimated and used to compute the network wide gevd based mwf however in this paper we aim to avoid centralizing the sensor signal observations in which case the network wide sensor signal correlation matrices cannot be estimated to this end we start from the so called distributed adaptive node specific signal estimation danse algorithm and include gevd based low rank approximations in the per node local computations remarkably the new algorithm is able to significantly compress the signal observations transmitted between the nodes while still converging to the network wide gevd based mwf as if each node would have access to all sensor signal observations even though the low rank approximations are applied locally at each node we provide a theoretical convergence analysis which shows that the algorithm converges to the network wide gevd based mwf under conditions that are less strict than in the original danse algorithm the convergence and performance of the algorithm are further investigated via numerical simulations

stochastic gradient based implementation of spatially preprocessed speech distortion weighted multichannel wiener filtering for noise reduction in hearing aids recently a generalized noise reduction scheme has been proposed called the spatially preprocessed speech distortion weighted multichannel wiener filter sp sdw mwf it encompasses the generalized sidelobe canceller gsc and a multichannel wiener filtering technique as extreme cases compared with the widely studied gsc with quadratic inequality constraint qic gsc the sp sdw mwf achieves a better noise reduction performance for a given maximum speech distortion level we develop a low cost stochastic gradient implementation of the sp sdw mwf to speed up convergence and reduce computational complexity the algorithm is implemented in the frequency domain experimental results with a behind the ear hearing aid show that the proposed frequency domain stochastic gradient algorithm preserves the benefit of the exact sp sdw mwf over the qic gsc while its computational cost is comparable to the least mean square based scaled projection algorithm for qic gsc

dual h _ infty algorithms for signal processing application to speech enhancement this paper deals with the joint signal and parameter estimation for linear state space models an efficient solution to this problem can be obtained by using a recursive instrumental variable technique based on two dual kalman filters in that case the driving process and the observation noise in the state space representation for each filter must be white with known variances these conditions however are too strong to be always satisfied in real cases to relax them we propose a new approach based on two dual hinfin filters once a new observation of the disturbed signal is available the first hinfin algorithm uses the latest estimated parameters to estimate the signal while the second hinfin algorithm uses the estimated signal to update the parameters in addition as the hinfin filter behavior depends on the choice of various weights we present a way to recursively tune them this approach is then studied in the following cases 1 consistent estimation of the ar parameters from noisy observations and 2 speech enhancement where no a priori model of the additive noise is required for the proposed approach in each case a comparative study with existing methods is carried out to analyze the relevance of our solution

multiple statistical models for soft decision in noisy speech enhancement most speech enhancement algorithms are based on the assumption that speech and noise are both gaussian in the discrete cosine transform dct domain for further enhancement of noisy speech in the dct domain we consider multiple statistical distributions i e gaussian laplacian and gamma as a set of candidates to model the noise and speech we first use the goodness of fit gof test in order to measure how far the assumed model deviate from the actual distribution for each dct component of noisy speech our evaluations illustrate that the best candidate is assigned to each frequency bin depending on the signal to noise ratio snr and the power spectral flatness measure psfm in particular since the psfm exhibits a strong relation with the best statistical fit we employ a simple recursive estimation of the psfm in the model selection the proposed speech enhancement algorithm employs a soft estimate of the speech absence probability sap separately for each frequency bin according to the selected distribution both objective and subjective tests are performed for the evaluation of the proposed algorithms on a large speech database for various snr values and types of background noise our evaluations show that the proposed soft decision scheme based on multiple statistical modeling or the psfm provides further speech quality enhancement compared with recent methods through a number of subjective and objective tests

on the design of pole zero approximations using a logarithmic error measure for obtaining a pole zero approximation of a linear discrete time system a new method is presented which minimizes the squared difference between the log magnitude spectrum of the system and that of the approximation using an iterative procedure a locally optimal solution is found for the poles and zeros of the system approximation

on the escape rate from a metastable state in a nonpotential system abstract the escape rate from a metastable state is computed for a nonpotential system described by a general multivariate fokker planck equation the result is a generalization of the escape rate obtained by other authors in the case of potential systems a straightforward application is found in the derivation of the grote hynes result for chemical rates in solution another characteristic application is the theory of nucleation in nonlinear chemical systems far from equilibrium which has been developed in a previous paper of the authors without a proper derivation of the nucleation rate formula j stat phys 50 1988 935

a convex concave problem with a nonlinear boundary condition in this paper we study the existence of nontrivial solutions of the problem u u u p 2uin u u q 2uon with 1 q 2 n 1 n 2 and 1 p 2n n 2 in the concave convex case i e 1 q 2 p if is small there exist two positive solutions while for large there is no positive solution when p is critical and q subcritical we obtain existence results using the concentration compactness method finally we apply the implicit function theorem to obtain solutions for small near u0 1

entropy extropy and information potential in stochastic systems far from equilibrium the relations between information entropy and energy which are well known in equilibrium thermodynamics are not clear far from equilibrium moreover the usual expression of the classical thermodynamic potentials is only valid near equilibrium in previous publications we showed for a chemical system maintained far from equilibrium that a new thermodynamic potential the information potential can be de4ned by using the stochastic formalism of the master equation here we extend this theory to a completely general discrete stochastic system for this purpose we use the concept of extropy which is de4ned in classical thermodynamics as the total entropy produced in a system and in the reservoirs during their equilibration we show that the statistical equivalent of the thermodynamic extropy is the relative information if a coarse grained description by means of the thermodynamic extensive variables is available which is the case for many macroscopic systems the coarse grained statistical extropy allows one to de4ne the information potential in the thermodynamic limit using this potential we study the evolution of such systems towards a non equilibrium stationary state we derive a general thermodynamic inequality between energy dissipation and information dissipation which sharpens the law of the maximum available work for non equilibrium systems c 2002 published by elsevier science b v

mean field self organized criticality the authors consider a variant of directed percolation in which the set of nodes is itself dependent on the progress of the percolation process for the mean field model studied the system is driven to its critical point from a broad range of initial conditions an advantage of the model is its explicit and easy solvability

nar estimators of spatial covariance matrices for adaptive array detection the theory of noise alone reference nar power estimation is extended to the estimation of spatial covariance matrices a nar covariance estimator insensitive to signal presence is derived the snr signal to noise ratio loss incurred by using this estimator is independent of the input snr and is less than that encountered with the maximum likelihood covariance estimator given that the same number of uncorrelated snapshots is available to both estimators the analysis assumes first a deterministic signal the results are extended and generalized to signals with unknown parameters or random signals for the random signal case generalized and quasi nar covariance estimators are presented

metastable lifetime for a class of chemical systems near criticality abstract we consider a n dimensional autocatalytic chemical system with a finite metastable state which tends to become critical away from the critical situation the relaxation time from the metastable state to the final absorbing state can be computed by using inverse of the stationary probability distribution of the system at its minimal value it is difficult to obtain explicit results on the stationary probability distribution in general for several variables nevertheless we can estimate this quantity near criticality and we show that its critical exponent is 2 under rather general conditions this result is verified explicitly on an example for which a direct calculation can be done

speech enhancement combining optimal smoothing and errors in variables identification of noisy ar processes in the framework of speech enhancement several parametric approaches based on an a priori model for a speech signal have been proposed when using an autoregressive ar model three issues must be addressed 1 how to deal with ar parameter estimation indeed due to additive noise the standard least squares criterion leads to biased estimates of ar parameters 2 can an estimation of the variance of the additive noise for each speech frame be obtained a voice activity detector is often used for its estimation 3 which estimation rules and techniques filtering smoothing etc can be considered to retrieve the speech signal our contribution in this paper is threefold first we propose to view the identification of the noisy ar process as an errors in variables problem this blind method has the advantage of providing accurate estimations of both the ar parameters and the variance of the additive noise second we propose an alternative algorithm to standard kalman smoothing based on a constrained minimum variance estimation procedure with a lower computational cost third the combination of these two steps is investigated it provides better results than some existing speech enhancement approaches in terms of signal to noise ratio snr segmental snr and informal subjective tests

speech enhancement using a minimum mean square error short time spectral amplitude estimator this paper focuses on the class of speech enhancement systems which capitalize on the major importance of the short time spectral amplitude stsa of the speech signal in its perception a system which utilizes a minimum mean square error mmse stsa estimator is proposed and then compared with other widely used systems which are based on wiener filtering and the spectral subtraction algorithm in this paper we derive the mmse stsa estimator based on modeling speech and noise spectral components as statistically independent gaussian random variables we analyze the performance of the proposed stsa estimator and compare it with a stsa estimator derived from the wiener estimator we also examine the mmse stsa estimator under uncertainty of signal presence in the noisy observations in constructing the enhanced signal the mmse stsa estimator is combined with the complex exponential of the noisy phase it is shown here that the latter is the mmse estimator of the complex exponential of the original phase which does not affect the stsa estimation the proposed approach results in a significant reduction of the noise and provides enhanced speech with colorless residual noise the complexity of the proposed algorithm is approximately that of other systems in the discussed class

minimal hypersurfaces and bordism of positive scalar curvature metrics let y g be a compact riemannian manifold of positive scalar curvature psc it is well known due to schoen yau that any closed stable minimal hypersurface of y also admits a psc metric we establish an analogous result for stable minimal hypersurfaces with free boundary furthermore we combine this result with tools from geometric measure theory and conformal geometry to study psc bordism for instance assume y 0 g 0 and y 1 g 1 are closed psc manifolds equipped with stable minimal hypersurfaces x 0 subset y 0 and x 1 subset y 1 under natural topological conditions we show that a psc bordism z bar g y 0 g 0 rightsquigarrow y 1 g 1 gives rise to a psc bordism between x 0 and x 1 equipped with the psc metrics given by the schoen yau construction

interaction of a hopf bifurcation and a symmetry breaking bifurcation stochastic potential and spatial correlations the multivariate master equation for a general reaction diffusion system is solved perturbatively in the stationary state in a range of parameters in which a symmetry breaking bifurcation and a hopf bifurcation occur simultaneously thestochastic potential u is in general not analytic however in the vicinity of the bifurcation point and under precise conditions on the kinetic constants it is possible to define a fourth order expansion ofu around the bifurcating fixed point under these conditions the domains of existence of different attractors including spatiotemporal structures as well as the spatial correlations of the fluctuations around these attractors are determined analytically the role of fluctuations in the existence and stability of the various patterns is pointed out

a new method of analysis of the effect of weak colored noise in nonlinear dynamical systems a systematic method for obtaining the asymptotic behavior of a dynamical system forced by colored noise in the limit of small intensity is developed it is based on the search of wkb solutions to the fokker planck equation for the joint probability density of the system and noise in which the perturbation expansion is continued to the first correction beyond the hamilton jacobi limit the method can be applied to noise with correlation time of order unity it is illustrated on the normal form of a pitchfork bifurcation where it is pointed out that additive noise can induce a shift of the most probable value this prediction is confirmed by numerical simulation of the stochastic differential equations

finite size scaling for mean field percolation by studying transfer matrix eigenvalues correlation lengths for a mean field directed percolation model are obtained both near and far from the critical regime near criticality finite size scaling behavior is derived and an analytic technique is provided for obtaining the finite size scaling function our methods involve the generating function matched asymptotic expansions and certain formulas developed for the study of eigenvalues of the transfer matrix for metastability

rank reduction for modeling stationary signals rank reduction is developed as a general principle for trading off model bias and model variance in the analysis and synthesis of signals the principle is applied to three basic problems stationary time series modeling stationary time series whitening and vector quantization each problem brings its own surprises and insights

physical interpretation of signal reconstruction from reduced rank matrices an enhanced version of a signal is obtained after subtraction of a linear combination of rank one matrices from the signal observation matrix this operation results in a nontoeplitz observation matrix by averaging the elements of this matrix a new toeplitz matrix is produced and a reconstructed signal is generated this averaging operation is examined and physical interpretations are given

quasipotentials for simple noisy maps with complicated dynamics the theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems leading to exponential estimates of invariant probabilities and mean first exit times in the present article we develop the mathematical foundation of this theory for discrete time systems following and extending the work of freidlin and wentzell and kifer we discuss strategies for calculating and estimating quasipotentials and show their application to one dimensionals unimodal maps the method proves to be especially suited for describing the noise scaling behavior of invariant probabilities e g for the map occurring as the limit of the feigenbaum period doubling sequence we show that the method allows statements about the scaling behavior in the case of localized noise too which does not originally lie within the scope of the quasipotential formalism

infinite range mean field percolation transfer matrix study of longitudinal correlation length two infinite range directed percolation models equivalent also to epidemic models are considered for a finite population finite number of sites n at each time directed axis step the general features of the transfer matrix spectrum evolution operator spectrum for the epidemic are studied numerically and compared with analytical predictions in the limitn one of the models is devised to allow numerical results to be obtained forn as high as nearly 800 for the largest longitudinal percolation correlation length relaxation time for epidemic the finite n behavior of this correlation length is studied in detail including scaling near the percolation transition exponential divergence withn above the percolation transition as well as other noncritical and critical point properties

the singular value decomposition and long and short spaces of noisy matrices using geometrical algebraic and statistical arguments it is clarified why and when the singular value decomposition is successful in so called subspace methods first the concepts of long and short spaces are introduced and a fundamental asymmetry in the consistency properties of the estimates is discussed the model which is associated with the short space can be estimated consistently but the estimates of the original data which follow from the long space are always inconsistent an expression is found for the asymptotic bias in terms of canonical angles which can be estimated from the data this allows all equivalent reconstructions of the original signals to be described as a matrix ball the center of which is the minimum variance estimate remarkably the canonical angles also appear in the optimal weighting that is used in weighted subspace fitting approaches the results are illustrated with a numerical simulation a number of examples are discussed

information potential and transition to criticality for certain two species chemical systems using the general results of the stochastic theory of chemical systems we consider a certain model of chemical reactions with two species for which it is possible to calculate the rst passage time explicitly and study the transition to criticality our method uses a non standard hamilton jacobi theory for the master equation introduced initially by kubo et al j stat phys 9 1973 51 which leads to solvable hamiltonian systems c 2000 elsevier science b v all rights reserved

addendum conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature is finite on the other hand the argument at the beginning of proposition 2 1 in e shows that if al b is finite then q m om is finite jin zhiren pointed out to me that the sobolev quotient q m o9m can be oo this is the case if we delete a small geodesic ball on a compact manifold without boundary with negative scalar curvature more generally the sobolev quotient is oo if the first eigenvalue for the conformal laplacian with respect to dirichlet boundary condition is negative in order to see that let p1 be the first eigenfunction for the problem

stochastic analysis of a hopf bifurcation master equation approach the effect of inhomogeneous fluctuations in a reaction diffusion system exhibiting a hopf bifurcation is analyzed using the master equation approach a taylor expansion of the logarithm of the stationary probability known as the stochastic potential is calculated this procedure displays marked analogies with the theory of normal forms the critical potential reduced to its local expansion around an arbitrary point of the limit cycle brings out the essential role played by the phase of the oscillating variables a comparison with the langevin analysis of walgraefet al j chem phys 78 6 3043 1983 is performed

data adaptive rank shaping methods for solving least squares problems there are two types of problems in the theory of least squares signal processing parameter estimation and signal extraction parameter estimation is called inversion and signal extraction is called filtering in this paper we present a unified theory of rank shaping for solving overdetermined and underdetermined versions of these problems we develop several data dependent rank shaping methods and evaluate their performance our key result is a data adaptive wiener filter that automatically adjusts its gains to accommodate realizations that are a priori unlikely the adaptive filter dramatically outperforms the wiener filter on a typical realizations and just slightly under performs it on typical realizations this is the most one can hope for in a data adaptive filter

equality governing nonequilibrium fluctuations and its information theory and thermodynamic interpretations the master equation is cast in the form of an equality involving the variation in the course of a transformation of a quantity playing the role of a generalized potential weighted with the probability of allowable transformations emanating from different initial states in the most general case this equality cannot be formulated entirely in terms of thermodynamic variables and state functions some conditions under which such a reduction becomes possible are identified and a comparison with fluctuation and work type relationships previously reported in the literature is carried out

stochastic analysis of symmetry breaking bifurcations master equation approach the multivariate master equation for a general reaction diffusion system is solved perturbatively in the vicinity of a bifurcation point leading to symmetry breaking transitions the possibility to express the result through a brazovskii type of potential is examined and a comparison with the langevin analysis of walgraefet al adv chem phys 49 311 1982 is performed

prewhitening for rank deficient noise in subspace methods for noise reduction a fundamental issue in connection with subspace methods for noise reduction is that the covariance matrix for the noise is required to have full rank in order for the prewhitening step to be defined however there are important cases where this requirement is not fulfilled e g when the noise has narrowband characteristics or in the case of tonal noise we extend the concept of prewhitening to include the case when the noise covariance matrix is rank deficient using a weighted pseudoinverse and the quotient singular value decomposition and we show how to formulate a general rank reduction algorithm that works also for rank deficient noise we also demonstrate how to formulate this algorithm by means of a quotient ulv decomposition which allows for faster computation and updating finally we apply our algorithm to a problem involving a speech signal contaminated by narrowband noise

a fast reduced rank interpolation method for prestack seismic volumes that depend on four spatial dimensions rank reduction strategies can be employed to attenuate noise and for prestack seismic data regularization we present a fast version of cadzow reduced rank reconstruction method cadzow reconstruction is implemented by embedding 4d spatial data into a level four block toeplitz matrix rank reduction of this matrix via the lanczos bidiagonalization algorithm is used to recover missing observations and to attenuate random noise the computational cost of the lanczos bidiagonalization is dominated by the cost of multiplying a level four block toeplitz matrix by a vector this is efficiently implemented via the 4d fast fourier transform the proposed algorithm significantly decreases the computational cost of rank reduction methods for multidimensional seismic data denoising and reconstruction synthetic and field prestack data examples are used to examine the effectiveness of the proposed method

dissipation of energy and of information in nonequilibrium reaction diffusion systems in this paper we prove a general inequality for the macroscopic relaxation towards a stationary nonequilibrium state namely the rate of dissipation of energy in the system always exceeds the rate of dissipation of information up to temperature here the information function is defined as the logarithm of the stationary probability distribution divided by the volume the equality is achieved if and only if the stationary probability distribution is an equilibrium distribution function this implies a lower bound for the work needed to produce a fluctuation away from the stationary state which should have experimental consequences

existence results for the p laplacian with nonlinear boundary conditions in this paper we study the existence of nontrivial solutions for the problem pu u p 2u in a bounded smooth domain rn with a nonlinear boundary condition given by u p 2 u f u on the boundary of the domain the proofs are based on variational and topological arguments

on the stationary distribution of self sustained oscillators around bifurcation points a double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones systems undergoing hopf bifurcations belong to this class as an illustrative example the generalized van der pol oscillator is considered around its bifurcation point a calculation is carried out up to third order in both the noise intensity and the bifurcation parameter damping coefficient

convergence of scalar flat metrics on manifolds with boundary under a yamabe type flow abstract we study a conformal flow for compact riemannian manifolds of dimension greater than two with boundary convergence to a scalar flat metric with constant mean curvature on the boundary is established in dimensions up to seven and in any dimensions if the manifold is spin or if it satisfies a generic condition

intrinsic fluctuations in chaotic dynamics abstract the multivariate master equation of a chemical system exhibiting spatio temporal chaos is investigated it is shown that the probability distribution in the presence of fluctuations remains centered on the underlying deterministic attractor

gsvd based optimal filtering for single and multimicrophone speech enhancement a generalized singular value decomposition gsvd based algorithm is proposed for enhancing multimicrophone speech signals degraded by additive colored noise this gsvd based multimicrophone algorithm can be considered to be an extension of the single microphone signal subspace algorithms for enhancing noisy speech signals and amounts to a specific optimal filtering problem when the desired response signal cannot be observed the optimal filter can be written as a function of the generalized singular vectors and singular values of a speech and noise data matrix a number of symmetry properties are derived for the single microphone and multimicrophone optimal filter which are valid for the white noise case as well as for the colored noise case in addition the averaging step of some single microphone signal subspace algorithms is examined leading to the conclusion that this averaging operation is unnecessary and even suboptimal for simple situations where we consider localized sources and no multipath propagation the gsvd based optimal filtering technique exhibits the spatial directivity pattern of a beamformer when comparing the noise reduction performance for realistic situations simulations show that the gsvd based optimal filtering technique has a better performance than standard fixed and adaptive beamforming techniques for all reverberation times and that it is more robust to deviations from the nominal situation as e g encountered in uncalibrated microphone arrays

fluctuations in mean field self organized criticality we present two models that exhibit self organized criticality at the mean field level these can be variously interpreted in epidemiological or chemical reaction terms by studying the master equation for these models we find however that only in one of them does the self organized critical behavior survive in the face of fluctuations for this model we show the spectrum of the evolution operator to have spectral collapse i e instead of a gap as would occur in noncritical behavior there are eigenvalues that approach zero as an inverse power of system size

fir filter representations of reduced rank noise reduction we show that the reduced rank output signal computed via truncated q svd is identical to that from an array of parallelly connected analysis synthesis finite impulse response fir filter pairs the filter coefficients are determined by the q svd and the filters provide an explicit description of the reduced rank noise reduction algorithm in the frequency domain

a subspace approach based on embedded prewhitening for voice activity detection this paper presents a subspace approach for voice activity detection vad the proposed approach is based on an embedded prewhitening scheme for the simultaneous diagonalization of the clean speech and noise covariance matrices to provide a decision rule based on likelihood ratio test in signal subspace domain experimental results show that the proposed subspace based vad algorithm outperforms the method using a gaussian model in a conventional discrete fourier transform domain at the low signal to noise conditions

variance analysis of linear simo models with spatially correlated noise in this paper we address the identification of linear time invariant single input multi output simo systems in particular we assess the performance of the prediction error method by quantifying

nonequilibrium potentials near instabilities the existence of polynomial approximations for nonequilibrium potentials determined by a master equation near an instability of arbitrary codimension with diagonalizable linear part is studied it is shown that the approximations exist provided some relations are satisfied between the coefficients of the master equation

lifetime for certain chemical systems at criticality abstract we consider an autocatalytic chemical system which is exactly at criticality so that the absorbing state lies on the boundary corresponding to the extinction of a species the average time to reach the absorbing state depends strongly on the distance from the initial state to the boundary and grows like the square root of the volume it can be estimated in generic critical situations these results are verified explicitly on an example for which a direct calculation can be done

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed

the behavior of the best sobolev trace constant and extremals in thin domains in this paper we study the asymptotic behavior of the best sobolev trace constant and extremals for the immersion w 1 p o l q o in a bounded smooth domain when it is contracted in one direction we find that the limit problem when rescaled in a suitable way is a sobolev type immersion in weighted spaces over a projection of o w 1 p p o a l q p o b for the special case p q this problem leads to an eigenvalue problem with a nonlinear boundary condition we also study the convergence of the eigenvalues and eigenvectors in this case