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Generalized eigenvector plays an essential role in the signal processing field. In this paper, we present a novel neural network learning algorithm for estimating the generalized eigenvector of a Hermitian matrix pencil. Differently from some traditional algorithms, which need to select the proper values of learning rates before using, the proposed algorithm does not need a learning rate and is very suitable for real applications. Through analyzing all of the equilibrium points, it is proven that if and only if the weight vector of the neural network is equal to the generalized eigenvector corresponding to the largest generalized eigenvalue of a Hermitian matrix pencil, the proposed algorithm reaches to convergence status. By using the deterministic discrete-time method, some convergence conditions, which can be satisfied with probability 1, are also obtained to guarantee its convergence. Simulation results show that the proposed algorithm has a fast convergence speed and good numerical stability. The real application demonstrates its effectiveness in tracking the optimal vector of beamforming.

This paper investigates robust downlink precoding for three-dimensional (3D) massive multi-input multi-output (MIMO) configuration with matrix manifold optimization. Starting with a posteriori channel model, we formulate the robust precoder design to maximize an upper bound of ergodic weighted sum-rate under a total power budget. We derive the generalized eigenvector structure for optimal precoder with matrix manifold optimization. However, since the precoding of multiple users is coupled in the structure, we maximize the objective function for each user in alternation and prove the solution of each individual problem is the generalized eigenvector corresponding to the maximum generalized eigenvalue. In accordance with this, we design an iterative algorithm and present its convergence analysis. Furthermore, we propose a Riemannian conjugate gradient (RCG) method to solve the generalized eigenvalue problem (GEP) for higher efficiency in the precoder design algorithm.

Given a self-adjoint operator A in a Hilbert space H, we analyze its spectral behavior when it is expressed in terms of generalized eigenvectors. Using the formalism of Gel’fand distribution bases, we explore the conditions for the generalized eigenspaces to be one-dimensional, i.e., for A to have a simple spectrum.

We prove a generalized Birman-Schwinger principle in the non-self-adjoint context. In particular, we provide a detailed discussion of geometric and algebraic multiplicities of eigenvalues of the basic operator of interest (e.g., a Schrodinger operator) and the associated Birman-Schwinger operator, and additionally offer a careful study of the associated Jordan chains of generalized eigenvectors of both operators. In the course of our analysis we also study algebraic and geometric multiplicities of zeros of strongly analytic operator-valued functions and the associated Jordan chains of generalized eigenvectors. We also relate algebraic multiplicities to the notion of the index of analytic operator-valued functions and derive a general Weinstein-Aronszajn formula for a pair of non-self-adjoint operators.

In this paper, we propose an iterative generalized eigenvector (GE) precoder design for three dimensional (3D) massive multi-input multi-output (MIMO) downlink with uniform planar array (UPA) and imperfect channel state information (CSI). We use a posterior beam based statistical channel model which includes the channel aging and the spatial correlation. We consider the problem of maximizing the expected sum-rate under a total power constraint. By replacing the expected sum-rate with their deterministic equivalents, we obtain the structure of the optimal linear precoders. The columns of the optimal precoding matrices are the generalized eigen-vectors of two matrices. One of the two matrices is related to the transmitted signal of the intended user, and the other is related to the power constraint and the leakage to other users. Furthermore, the two matrices are also functions of the precoders, thus iterative updates of the precoder are needed. Then, we investigate the power allocation and the computation of the Lagrangian multiplier, and propose an iterative algorithm based on the structure of the optimal precoders. Simulation results show that the proposed precoders can achieve significantly performance gain than the widely used regularized zero forcing (RZF) precoder and signal to leakage plus noise ratio (SLNR) precoder.

Generalized eigenvectors are key tools in the theory of rigged Hilbert spaces. Let H be a Hilbert space and let Φ be a dense subspace of H . Let A be a densely defined linear operator in H such that Φ ⊂ DA and AΦ ⊂ Φ. The generalized eigenvectors of A are the eigenvectors of the algebraic dual of A |Φ. In the case where Φ is endowed with a topology τ finer than the norm topology inherited from H , generalized eigenvectors that are τ-continuous may be of great interest. We discuss conditions which ensure the existence of representations associated to generalized eigenvectors of A. As an application, we review and refine Bohm's study of the algebra of the quantum harmonic oscillator.

V diplomski nalogi je formuliran algoritem za iskanje lastnih vrednosti in lastnih vektorjev brez uporabe determinante. Za algoritem je ključno razumevanje linearne neodvisnosti oziroma odvisnosti, zato je v delu to temeljito opisano. Definirali smo lastne vrednosti, lastne vektorje, matrični polinom, minimalni polinom matrike, minimalni polinom vektorja glede na matriko in v povezavi s temi pojmi navedli trditve, ki so nam pomagale pri konstrukciji algoritma. Postopek za iskanje lastnih vrednosti in vektorjev smo skozi delo gradili postopoma. Najprej smo ga uporabili na nedefektnih matrikah. Nato smo si pogledali še definicijo defektnih matrik, korenskih lastnih vektorjev, Jordanovo verigo korenskih lastnih vektorjev in trditve v povezavi z njimi. Skozi celotno diplomsko nalogo so nova dognanja uporabljena na primerih. Na koncu smo zapisali celoten univerzalen algoritem, ne glede na začetno matriko. In this bachelor thesis we formulate the algorithm for finding eigenvalues and eigenvectors without the use of determinant. For the algorithm to work, the understanding of linear independance and dependance is crucial, that is why we chose to present these two principles in more detail. We defined eigenvalues, eigenvectors, matrix polynomial, minimal polynomial of the matrix, and minimal polynomial of a vector with respect to matrix. These definitions and theorems helped us to construct our algorithm. We built our method step-by-step through our bachelor thesis. First, we used it on non defective matrices. Then we defined defective matrices, generalized vectors and Jordan chain of generalized eigenvectors. Throughout the thesis, examples are used to show what we have discovered till then. In the end, we formulated the whole universal algorithm, which works no matter what kind of the matrix we start with.

This article is concerned with the identification of autoregressive with exogenous inputs (ARX) models. Most of the existing approaches like prediction error minimization and state-space framework are widely accepted and utilized for the estimation of ARX models but are known to deliver unbiased and consistent parameter estimates for a correctly supplied guess of input-output orders and delay. In this paper, we propose a novel automated framework which recovers orders, delay, output noise distribution along with parameter estimates. The primary tool utilized in the proposed framework is generalized spectral decomposition. The proposed algorithm systematically estimates all the parameters in two steps. The first step utilizes estimates of the order by examining the generalized eigenvalues, and the second step estimates the parameter from the generalized eigenvectors. Simulation studies are presented to demonstrate the efficacy of the proposed method and are observed to deliver consistent estimates even at low signal to noise ratio (SNR)., 8 pages, accepted at MTNS 2020

The symplectic approach is used to establish the unified framework for solving governing equations of some thin plate problems in elasticity. By introducing appropriate functions, the given partial differential equation is first transferred into a separable Hamiltonian system. The completeness of the system of generalized eigenvectors of corresponding Hamiltonian operator matrix is proved, which serves as the theoretical foundation of symplectic approach. Utilizing the expansion theorems, the general solutions of the boundary value problem for partial differential equation under consideration are obtained. Moreover, the bending, buckling, and free vibration problems of fully clamped rectangular thin plates governed by the given partial differential equation are solved analytically by the technique of superposition. Numerical results for bending, buckling, and free vibration plates are presented to demonstrate the availability and validity of the approach by comparison with those available in the literatures.

Generalized principal component analysis (GPCA) has been an active area of research in statistical signal processing for decades. It is used, e.g., for denoising in subspace tracking as the noise of different nature is incorporated into the procedure of maximizing signal-to-noise ratio (SNR). This paper presents a fixed-point approach concerning the principal generalized eigenvector extraction. It is based on the basis iteration for maximizing the generalized Rayleigh quotient (GRQ) with a given matrix pencil. The proposed approach extracts multiple generalized eigenvectors of a matrix pencil by exploiting the orthogonal complement structure of its estimation. It has no requirement to choose the commonly used step size. This enhances its practical applicability, as selecting an appropriate step size is a bottleneck for most gradient flow based algorithms. Our approach is more suitable for online processing because of its easy implementation and low computational complexity. To show the efficacy, efficiency and practical applicability of the proposed algorithm, we conduct several experiments, two of which concern smart antenna and blind source separation applications. Our simulation results show that the proposed algorithm outperforms several existing algorithms in terms of convergence speed as well as computational time.

The variable span linear filters (VSLFs) constitute a unified framework of conventional subspace and linear filtering techniques for noise reduction. The construction of VSLFs, however, relies on the generalized eigendecomposition (GEVD) methods, which are computationally expensive. This in turn stymies the employment of such filters in practical online processing problems. To address this issue, we first propose in this paper a fixed-point iteration technique to extract the generalized eigenvectors. It is based on maximizing the pre-whitened generalized Rayleigh quotient (GRQ). We then integrate this technique with online statistic estimation to construct VSLFs. Our proposed method is computationally efficient and can also harness parallel architectures. To show its effectiveness, we consider a speech enhancement application and compare the results with those of several existing methods.

In this paper, we study spectral decompositions with respect to a system of generalized eigenvectors of second-order differential operators on an interval whose resolvents possess convolution representations. We obtain the convolution representation of resolvents of second-order differential operators on an interval with integral boundary conditions. Then, using the convolution generated by the initial differential operator, we construct the Fourier transform. A connection between the convolution operation in the original functional space and the multiplication operation in the space of Fourier transforms is established. Finally, the problem on the convergence of spectral expansions generated by the original differential operator is studied. Examples of convolutions generated by operators are also presented.

We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer theorem for building irreducible representations, the McKay correspondence, and Pitman's 2 M − X theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analysis using generalized eigenvectors proves successful.

Dynamic-inner canonical correlation analysis (DiCCA) extracts dynamic latent variables from high-dimensional time series data with a descending order of predictability in terms of $R^2$ . The reduced dimensional latent variables with rank-ordered predictability capture the dynamic features in the data, leading to easy interpretation and visualization. In this article, numerically efficient algorithms for DiCCA are developed to extract dynamic latent components from high-dimensional time series data. The numerically improved DiCCA algorithms avoid repeatedly inverting a covariance matrix inside the iteration loop of the numerical DiCCA algorithms. A further improvement using singular value decomposition converts the generalized eigenvector problem into a standard eigenvector problem for the DiCCA solution. Another improvement in model efficiency in this article is the dynamic model compaction of the extracted latent scores using autoregressive integrated moving average (ARIMA) models. Integrating factors, if existed in the latent variable scores, are made explicit in the ARIMA models. Numerical tests on two industrial datasets are provided to illustrate the improvements.

This paper presents an l1-penalized extension of the adaptive normalized quasi-Newton algorithm (Nguyen and Yamada, 2013) which was established for online generalized eigenvalue problem. The proposed extension aims to exploit effectively the sparsity as a priori knowledge for efficient subspace tracking in signal processing and is also motivated by recent sparsity-aware eigenvector analysis in data sciences, e.g., sparse principal component analysis. For such an extension, we newly introduce l1 penalty into a non-convex criterion which has been used to characterize, as its stationary point, the generalized eigen-pair. The proposed subspace tracking algorithm is derived by applying a quasi-Newton type step to the new criterion followed by a normalization step. A convergence analysis is given in the case for decaying weight of the penalty. We also discuss potential applications, e.g., online sparse principal component analysis, by controlling the weight sequence of the l1 penalty. Numerical experiments demonstrate that the proposed algorithm (i) can improve the subspace tracking performance even for noisy observation of random vectors whose covariance matrix pencil has sparse principal generalized eigenvector and (ii) can promote the interpretability of the estimate of the principal generalized eigenvector.

Generalized eigenvalue decomposition has many advantages when it is applied in modern signal processing. Compared with other methods, neural network model-based algorithms provide an efficient way to solve such problems online. Generalized feature extraction algorithms based on neural network models have been described in the literature. However, the majority of the existing algorithms can only extract the principal generalized eigenvector(s) or eigensubspace. To extract principal and minor generalized eigenvectors from two vector sequences, in this paper, two different information criteria are proposed, and a unified algorithm for the extraction of multiple components in a parallel way by simply altering the sign is derived based on these information criteria, which is feasible for generalized principal and minor component analysis. Moreover, all the corresponding principal and minor generalized eigenvalues can be extracted simultaneously because the desired equilibrium point depends on these values. Thus, the proposed algorithm can perform multiple generalized eigenpair extraction. The proposed algorithm possesses four properties: unification, self-stability, parallel extraction and generalized eigenpair extraction, that few of the existing algorithms can encompass. The global convergence and self-stability property of the proposed algorithm are proved through the Lyapunov method and ordinary differential equation method, respectively. The proposed algorithm has a fast convergence speed, high precision and strong tracking ability. Finally, numerical examples and applications are explored to further demonstrate the efficiency of the proposed algorithm.

In this paper, we investigate a min-plus analogue of Jordan canonical forms of matrices. We first define the generalized eigenvector of a min-plus matrix A as an eigenvector of the kth power of A f...

${;M_v};-$matrix is a matrix of the form $A = sI-B$, where $ 0 \le \rho (B) \le s$ and $B$ is an eventually nonnegative matrix. In this paper, $M_v-$matrices concerning the Perron-Frobenius theory are studied. Specifically, sufficient and necessary conditions for an $M_v-$matrix to have positive left and right eigenvectors corresponding to its eigenvalue with smallest real part without considering or not if $index_{;0}; B \leq 1$ are stated and proven. Moreover, analogous conditions for eventually nonnegative matrices or $M_v-$matrices to have all the non Perron eigenvectors or generalized eigenvectors not being nonnegative are studied. Then, equivalent properties of eventually exponentially nonnegative matrices and $M_v-$matrices are presented. Various numerical examples are given to support our theoretical findings.

This paper shows that the generalized eigenvalues of a symmetric positive matrix pair can be computed efficiently under more general hypothesises by the proposed recurrent neural network (RNN) in Liu et al. (2008). More precisely, it is proved that based on more general hypothesises, the state solution of the proposed RNN converges to the generalized eigenvector of symmetric positive pair, and its related generalized eigenvalue depends on the initial point of the state solution. Furthermore, the related largest and smallest generalized eigenvalues can also be obtained by the proposed RNN. Some related numerical experiments are also presented to illustrate our results.

In this article, a class of upwind schemes is proposed for systems, each of which yields an incomplete set of linearly independent eigenvectors. The theory of Jordan canonical forms is used to complete such sets through the addition of generalized eigenvectors. A modified Burgers’ system and its extensions generate δ,δ′, δ″,⋯,δn waves as solutions. The performance of flux difference splitting-based numerical schemes is examined by considering various numerical examples. Since the flux Jacobian matrix of pressureless gas dynamics system also produces an incomplete set of linearly independent eigenvectors, a similar framework is adopted to construct a numerical algorithm for a pressureless gas dynamics system.

A problem of characterizing conditions under which a topological change in a network of differential–algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topological change. A condition in terms of eigenvectors of the nominal system is derived to check for existence of possibly indiscernible topological changes. For homogeneous networks this condition simplifies to the existence of an eigenvector of the Laplacian of network having equal components. Lastly, a rank condition is derived which can be used to check if a topological change preserves regularity of the nominal network.

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein-Uhlenbeck systems $(X^\varepsilon_t(x))_{t\geqslant 0}$ with $\varepsilon$-small additive L\'evy noise and initial value $x$. The driving noise processes include Brownian motion, $\alpha$-stable L\'evy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp $\infty/0$-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $\mu^\varepsilon$ along a time window centered on a precise $\varepsilon$- and $x$-dependent time scale $t_\varepsilon^x$. In many interesting situations such as reversible (L\'evy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data $x$ we obtain the stronger result $\mathcal{W}_p(X^\varepsilon_{t_\varepsilon + r}(x), \mu^\varepsilon) \cdot \varepsilon^{-1} \rightarrow K\cdot e^{-q r}$ as $\varepsilon \rightarrow 0$ for any $r\in \mathbb{R}$, some spectral constants $K, q>0$ and any $p\geqslant 1$ whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of $\mathcal{Q}$. Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to $\varepsilon$-small Brownian motion or $\alpha$-stable L\'evy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature., Comment: 47 pages

We connect the Grover walk with sinks to the Grover walk with tails. The survival probability of the Grover walk with sinks in the long time limit is characterized by the centered generalized eigenspace of the Grover walk with tails. The centered eigenspace of the Grover walk is the attractor eigenspace of the Grover walk with sinks. It is described by the persistent eigenspace of the underlying random walk whose support has no overlap to the boundaries of the graph and combinatorial flow in the graph theory., 26 pages; 4 figures

Remote photoplethysmography (rPPG) has been at the forefront recently, thanks to its capacity in estimating non-contact physiological parameters such as heart rate and heart rate variability (Wang et al. in FBB 6:33, 2018). rPPG signals are typically extracted from facial videos by performing spatial averaging to obtain temporal RGB traces. Although this spatial averaging simplifies computation, it is accompanied by loss of essential spatial information which might reveal interesting relationships between signals from different spatial regions. In this article, we present a novel algorithm adapted from generalized eigenvalue decomposition (GEVD) to estimate this spatial rPPG distribution. GEVD is an extremely versatile algorithm that finds uses in signal and image processing and analytical problems such as principal component analysis and Fisher discriminant analysis (Ghojogh et al. in Tutorial 2: 1–8, 2019)(Han and Clemmensen in PR 49:43-54, 2016). It is performed using the QZ algorithm (Moler and Stewart in JNA 10(2):241–256, 2010), which in turn uses Householder transformations (Householder in JACM 5(4):339–342, 1958) to extract generalized eigenvectors of a pair of matrices. We adapt the QZ algorithm for the domain of spatio-temporal biomedical signals such as remote photoplethysmography (rPPG), electrocardiography and electroencephalography signals. We call this algorithm Temporal-QZ, which employs vectorization techniques to extract generalized eigenvectors over spatial data points simultaneously. We validate this extension in the domain of remote photoplethysmography (rPPG) measurement, for the estimation of spatial rPPG distribution of skin.

The aim of this paper is to give a precise proof of the completeness of Lamb modes and associated modes. This proof is relatively simple and short but relies on two powerful mathematical theorems. The first one is a theorem on elliptic systems with a parameter due to Agranovich and Vishik. The second one is a theorem due to Locker which gives a criterion to show the completeness of the set of generalized eigenvectors of a Hilbert-Schmidt discrete operator., 19 pages

International audience; The generalized Hermitian eigendecomposition problem is ubiquitous in signal and machine learning applications. Considering the need of processing streaming data in practice and restrictions of existing methods, this paper is concerned with fast and efficient generalized eigenvectors tracking. We first present a computationally efficient algorithm based on randomization termed alternate-projections randomized eigenvalue decomposition (APR-EVD) to solve a standard eigenvalue problem. By exploiting rank-1 strategy, two online algorithms based on APR-EVD are developed for the dominant generalized eigenvectors extraction. Numerical examples show the practical applicability and efficacy of the proposed online algorithms.

For a class of linear time-delay control systems satisfying the property of completability of the generalized eigenvectors, we prove that the problems of complete stabilizability and exact null controllability are equivalent.

In this chapter, we recall some facts related to the evolutionary problem such as Hille-Yosida theorem, differentiability of the semigroup. We also give some properties of fractional operators and expansion of solution on generalized eigenvectors of operators in Hilbert space. In this chapter some elementary properties of semigroups are given.

This chapter concentrate on a selection of applications in mathematical physics and mechanics to which the results of the preceding chapters are applied. This chapter contains some applications in mathematical physics and mechanics to investigate the expansion of solution in terms of generalized eigenvectors for a rectilinear transport equation and the behavior of the resolvent in the case of the Lame system.

Spectral Regression (SR) is a novel dimensionality reduction framework for efficient regularized subspace learning, which is fundamentally based on regression and spectral graph analysis. In this paper, we present a quantum algorithm for SR, where quantum Gram-Schmidt process are proposed and quantum singular value estimation (SVE) technique is applied for regression problem. The quantum algorithm involves two core phases: (1) With labels of the data set, we present a quantum Gram-Schmidt process subroutine based on the quantum block-encoding technique for generalized eigenvectors. (2) A quantum method for ridge regression is presented based on SVE, where the extended Hermitian form of the data matrix is not required. It is shown that our quantum algorithm can achieve an exponential speedup over the classical counterpart for the data matrix with well condition number and small sample size.

Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold.

A new computational strategy for the elasto-plastic analysis of framed structures is proposed. The approach is composed of two levels of analysis: the frame level, based on a 3D mixed beam element, and the cross-section level, described through the Generalized Eigenvectors approach whose formulation is here extended for the analysis of materials with elasto-plastic behaviour. The proposed method, suitable for analyzing single beams as well as assemblages, can tackle composite cross-sections by taking into account all the stress components of the 3D solution.

In this paper, we study a class of linear evolution equations of fractional order that are degenerate on the kernel of the operator under the sign of the derivative and on its relatively generalized eigenvectors. We prove that in the case considered, in contrast to the case of first-order degenerate equations and equations of fractional order with weak degeneration (i.e., degeneration only on the kernel of the operator under the sign of the derivative), the family of analytical in a sector operators does not vanish on relative generalized eigenspaces of the operator under the sign of the derivative, has a singularity at zero, and hence does not determine any solution of a strongly degenerate equation of fractional order. For the case of a strongly degenerate equation of integer order this fact does not hold, but the behavior of the family of resolving operators at zero cannot be examined by ordinary method.

Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors. We analyze here what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks.

In this paper, we develop a new deflationary blind source extraction (BSE) algorithm that extracts source signals in a sequential fashion via the joint generalized eigenvectors of a set of covariance matrix pencils. The new concept of joint generalized eigenvector is defined. We prove that these vectors can be made unique and identical to the source extraction vectors with properly selected matrix pencils. To resolve the open problem of estimating joint generalized eigenvectors, we develop an approach based on the deflation operation and the proportional property of the joint generalized eigenvectors. Specifically, with the proportional property, we show that the estimation problem can be formulated as an optimization involving a quadratic cost function and a unit-rank matrix constraint. An efficient iterative algorithm is then developed by applying the gradient search, matrix shrinkage, deflation, and symmetry-preserving vectorization techniques. This algorithm estimates the joint generalized eigenvectors and conducts BSE sequentially. Its computational complexity and convergence are analyzed. Simulations demonstrate that this algorithm outperforms many typical BSE or blind source separation algorithms. In particular, the new algorithm is more robust to both heavy noise and ill-conditioned mixing matrices.

We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous. Uniform asymptotics of generalised eigenvectors and conditions implying complete indeterminacy are also provided., 27 pages

An improved blind channel equalization method is proposed based on the generalized eigenvector algorithm (GEA) in this paper. This new method can blindly equalize the desired signal in the presence of strong co-channel interference. The basic idea underlying the improved GEA method is that higher-order cumulants can be sensitive to frequency offset. By exploiting this property of higher-order statistics, blind equalizer can be designed to equalize the desired signal with known frequency offset while suppressing the interference with a different frequency offset. Simulation results are shown to demonstrate the effectiveness of the proposed method.

3rd International Conference on Analysis and Applied Mathematics (ICAAM) -- SEP 07-10, 2016 -- Almaty, KAZAKHSTAN WOS:000439421100020 We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular, it is shown that the problem under consideration has precisely denumerable many eigenvalues lambda(1),lambda(2),..., which are real and tends to +infinity. Moreover, it is proven that the generalized eigenvectors form a Riesz basis of the adequate Hilbert space. Inst Math & Math Modeling, Al Farabi Kazakh Natl Univ, L N Gumilyov Eurasian Natl Univ