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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.

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.

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.

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

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.

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.

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

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.

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.

We show that it is possible for every non-diagonalizable stochastic 3 × 3 matrix to be perturbed into a diagonalizable stochastic matrix with the eigenvalues, arbitrarily close to the eigenvalues of the original matrix, with the same principal eigenspaces. An algorithm is presented to determine a perturbation matrix, which preserves these spectral properties. Additionally, a relation is proved between the eigenvectors and generalized eigenvectors of the original matrix and the perturbed matrix.

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.

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

For adaptive extraction of generalized eigensubspace, Nguyen, Takahashi and Yamada proposed a scheme for solving generalized Hermitian eigenvalue problem based on nested orthogonal complement structure. This scheme can extract multiple generalized eigenvectors by combining with any algorithm designed for estimation of the first minor generalized eigenvector. In this paper, we carefully analyse the effect of a discontinuous function employed in the scheme, and show that the discontinuous function can cause unsmooth changes of the estimates by the scheme in its adaptive implementation. To remedy the weakness, we newly introduce a projection step, for smoothing, without increasing the order of the computational complexity. Numerical experiments show that the learning curves of the non-first generalized eigenvectors are improved drastically through the proposed smoothing even when the original scheme results in unexpected performance degradation.

The Eigendecomposition of quadratic forms (symmetric matrices) guaranteed by the spectral theorem is a foundational result in applied mathematics. Motivated by a shared structure found in inferential problems of recent interest---namely orthogonal tensor decompositions, Independent Component Analysis (ICA), topic models, spectral clustering, and Gaussian mixture learning---we generalize the eigendecomposition from quadratic forms to a broad class of "orthogonally decomposable" functions. We identify a key role of convexity in our extension, and we generalize two traditional characterizations of eigenvectors: First, the eigenvectors of a quadratic form arise from the optima structure of the quadratic form on the sphere. Second, the eigenvectors are the fixed points of the power iteration. In our setting, we consider a simple first order generalization of the power method which we call gradient iteration. It leads to efficient and easily implementable methods for basis recovery. It includes influential Machine Learning methods such as cumulant-based FastICA and the tensor power iteration for orthogonally decomposable tensors as special cases. We provide a complete theoretical analysis of gradient iteration using the structure theory of discrete dynamical systems to show almost sure convergence and fast (super-linear) convergence rates. The analysis also extends to the case when the observed function is only approximately orthogonally decomposable, with bounds that are polynomial in dimension and other relevant parameters, such as perturbation size. Our perturbation results can be considered as a non-linear version of the classical Davis-Kahan theorem for perturbations of eigenvectors of symmetric matrices., 69 pages

Purpose To introduce new solution methods for the Bloch and Bloch-McConnell equations and compare them quantitatively to different known approaches. Theory and Methods A new exact solution per time step is derived by means of eigenvalues and generalized eigenvectors. Fast numerical solution methods based on asymmetric and symmetric operator splitting, which are already known for the Bloch equations, are extended to the Bloch-McConnell equations. Those methods are compared to other numerical methods including spin domain, one-step and multi-step methods, and matrix exponential. Error metrics are introduced based on the exact solution method, which allows to assess the accuracy of each solution method quantitatively for arbitrary example data. Results Accuracy and performance properties for nine different solution methods are analyzed and compared in extensive numerical experiments including various examples for non-selective and slice-selective MR imaging applications. The accuracy of the methods heavily varies, in particular for short relaxation times and long pulse durations. Conclusion In absence of relaxation effects, the numerical results confirm the rotation matrices approach as accurate and computationally efficient Bloch solution method. Otherwise, as well as for the Bloch-McConnell equations, symmetric operator splitting methods are recommended due to their excellent numerical accuracy paired with efficient run time.

We pose a linear matrix differential-algebraic boundary-value problem generalizing the traditional linear boundary-value problems for differential-algebraic equations. We have found the constructive conditions of existence and an algorithm of construction of the solutions of a linear matrix differentialalgebraic boundary-value problem. We propose the construction of a generalized Green operator for the determination of solutions of the linear differential-algebraic boundary-value problem and give some examples of construction of such solutions.

A new method is proposed to compute the eigenvector derivative of a quadratic eigenvalue problem (QEP) analytically dependent on a parameter. It avoids the linearization of the QEP. The proposed method can be seen as an improved incomplete modal method. Only a few eigenvectors of the QEP are required. The contributions of other eigenvectors to the desired eigenvector derivative are obtained by an iterative scheme. From this point of view, our method also can be seen as an iterative method. The convergence properties of the proposed method are analyzed. The techniques to accelerate the proposed method are provided. A strategy is developed for simultaneously computing several eigenvector derivatives by the proposed method. Finally some numerical examples are given to demonstrate the efficiency of our method.

In this paper we consider the problem of computing generalized eigenvectors of a matrix pencil in real Schur form. In exact arithmetic, this problem can be solved using substitution. In practice, substitution is vulnerable to floating-point overflow. The robust solvers xtgevc in LAPACK prevent overflow by dynamically scaling the eigenvectors.These subroutines are scalar and sequential codes which compute theeigenvectors one by one. In this paper, we discuss how to derive robust algorithms which are blocked and parallel. The new StarNEig librarycontains a robust task-parallel solver Zazamoukh which runs on top of StarPU. Our numerical experiments show that Zazamoukh achieves a super-linear speedup compared with dtgevc for sufficiently large matrices. NLAFET

In this paper we give necessary and sufficient conditions for a bounded linear Hilbert space operator to be an m-isometry for an unspecified m written in terms of conditions that are applied to “one vector at a time”. We provide criteria for orthogonality of generalized eigenvectors of an (a priori unbounded) linear operator T on an inner product space that correspond to distinct eigenvalues of modulus 1. We also discuss a similar question of when Jordan blocks of T corresponding to distinct eigenvalues of modulus 1 are “orthogonal”.

In [M. Walter et al., Science 340, 1205, 7 June (2013)], they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification via polytopes and the eigenvalues of the single-particle states. In this paper, for $4n$ qubits, we show the invariance of algebraic multiplicities (AMs) and geometric multiplicities (GMs) of eigenvalues and the invariance of sizes of Jordan blocks (JBs) of the coefficient matrices under SLOCC. We explore properties of spectra, eigenvectors, generalized eigenvectors, standard Jordan normal forms (SJNFs), and Jordan chains of the coefficient matrices. The properties and invariance permit a reduction of SLOCC classification of $4n$ qubits to integer partitions (in number theory) of the number $2^{2n}-k$ and the AMs., 15 pages, published version

Let p be a rational prime, v p the normalized p-adic valuation on Z , q > 1 a power of p and A = F q [ t ] . Let ℘ ∈ A be an irreducible polynomial and n ∈ A a non-zero element which is prime to ℘. Let k ≥ 2 and r ≥ 1 be integers. We denote by S k ( Γ 1 ( n ℘ r ) ) the space of Drinfeld cuspforms of level Γ 1 ( n ℘ r ) and weight k for F q ( t ) . Let n ≥ 1 be an integer and a ≥ 0 a rational number. Suppose that n ℘ has an irreducible factor of degree one and the generalized eigenspace in S k ( Γ 1 ( n ℘ r ) ) of slope a is one-dimensional. In this paper, under an assumption that a is sufficiently small, we construct a family { F k ′ | v p ( k ′ − k ) ≥ log p ⁡ ( p n + a ) } of Hecke eigenforms F k ′ ∈ S k ′ ( Γ 1 ( n ℘ r ) ) of slope a such that, for any Q ∈ A , the Hecke eigenvalues of F k and F k ′ at Q are congruent modulo ℘ κ with some κ > p v p ( k ′ − k ) − p n − a .

We discuss the description of eigenspace of a quantum walk model $U$ with an associating linear operator $T$ in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of $U$ without the spectral decomposition of $T$. Arguments in this direction reveal the eigenspaces of $U$ characterized by the generalized kernels of linear operators given by $T$., Comment: 17 pages

The conditions of regularization and the structure of generalized Green’s operator for a re-gularized linear matrix differential-algebraic boundary-value problem are found. To solve the problem of regularization of a generalized matrix differential-algebraic boundary-value problem, the original conditions of solvability and the structure of the general solution of a matrix equation of the Sylvester type are used.

We study the pathology that causes tropical eigenspaces of distinct supertropical eigenvalues of a nonsingular matrix $A$, to be dependent. We show that in lower dimensions the eigenvectors of distinct eigenvalues are independent, as desired. The index set that differentiates between subsequent essential monomials of the characteristic polynomial, yields an eigenvalue $\lambda$, and corresponds to the columns of the eigenmatrix $A+\lambda I$ from which the eigenvectors are taken. We ascertain the cause for failure in higher dimensions, and prove that independence of the eigenvectors is recovered in case a certain "difference criterion" holds, defined in terms of disjoint differences between index sets of subsequent coefficients. We conclude by considering the eigenvectors of the matrix $A^\nabla : = \det(A)^{-1}\adj(A)$ and the connection of the independence question to generalized eigenvectors., Comment: The first author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowship

For generic values of q, all the eigenvectors of the transfer matrix of the U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N., 50pp, 2 figures, v2: few typos are fixed, Nucl. Phys. B (2016)

We find necessary and sufficient conditions for solvability and the construction of the generalized Green operator for Noetherian linear boundary-value problem for a linear matrix differential equation. We propose an operator, which leads a linear matrix algebraic equation to the traditional linear algebraic system with a rectangular matrix. We use pseudoinverseMoore–Penrose matrices and orthogonal projections for solving a linear algebraic system.

We use the left vanishing eigenvector to prove various well-known conditions for determining the nonsingularity of matrices via row sums. This is in contrast to the classical approach of using the right vanishing eigenvector. We show that on occasion this approach results in simpler proofs and generalizations of well-known results. We also present a simple proof of a generalized Gudkov's theorem.

The generalized Hermitian eigenvalue problem (GHEP) is of great use in modern signal processing. Compared with other methods, neural network model based algorithms provide an efficient way to solve such problems online. Up to now, a class of neural network model based generalized feature extraction algorithms have been reported in the literature. However, most existing noncoupled algorithms suffer from the so-called speed-stability problem. In contrast, coupled learning algorithms estimate the generalized eigenvector and generalized eigenvalue simultaneously in coupled manners. Therefore they can solve the speed-stability problem and perform better than noncoupled algorithms. In this paper, we first propose a coupled generalized system, which is obtained by using the Newton’s method and a novel generalized information criterion. Then, based on this coupled generalized system, we successfully obtain two coupled algorithms with normalization steps for minor/principal generalized eigen-pairs extraction. And the technique of multiple generalized eigen-pairs extraction is also introduced in this paper. The convergence of the proposed algorithms is justified by deterministic discrete-time (DDT) approach, with the efficiency and feasibility validated by experimental results as well. Compared with similar work, the proposed algorithms have lower computational complexity, better numerical stability and higher estimation accuracy.

We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.

In this study, convection-pressure split Euler flux functions which contain weakly hyperbolic convective subsystems are analyzed. A system of first-order partial differential equations (PDEs) is said to be weakly hyperbolic if the corresponding flux Jacobian does not contain a complete set of linearly independent (LI) eigenvectors. Thus, the application of existing flux difference splitting (FDS) based schemes, which depend heavily on both eigenvalues and eigenvectors, are non-trivial to such systems. In the case of weakly hyperbolic systems, a required set of LI eigenvectors can be constructed through the addition of generalized eigenvectors by utilizing the theory of Jordan canonical forms. Once this is achieved for a weakly hyperbolic convective subsystem, an upwind solver can be constructed in the splitting framework. In the present work, the above approach is used for developing two new schemes. The first scheme is based on the Zha–Bilgen type splitting while the second is based on the Toro–Vazquez splitting. Both the schemes are tested on various benchmark problems in one-dimension (1-D) and two-dimensions (2-D). The concept of generalized eigenvectors based on Jordan forms is found to be useful in dealing with the weakly hyperbolic parts of the considered Euler systems.

Regarding the eventuality property of a discrete-time affine system (DTAS), we investigate whether all trajectories in the system leave a given region. Based on the compactness of the region, this study is divided into two cases. For a compact region, we give an exact characterisation of the eventuality property. For a not necessarily compact region, we argue the verification problem can be converted to the evaluation of an invariant cone under a linear map. Finally we provide a sufficient condition for the eventuality property based on the knowledge of eigenvalues and generalized eigenvectors of a matrix. A computational evaluation of the analytical criteria is outlined.

This paper provides decay bounds for Green matrices and generalized eigenvectors of block Jacobi operators when the real part of the spectral parameter lies in a bounded gap of the operator’s essential spectrum. The case of the spectral parameter being an eigenvalue is also considered. It is also shown that if the matrix entries commute, then the estimates can be refined. Finally, various examples of block Jacobi operators are given to illustrate the results.

The goal of this paper is to establish a widely applicable method for exploiting the sparsity in generalized eigenvector estimation. We propose an $\ell _{1}$-penalized extension of the Adaptive normalized quasi-Newton algorithm (Nguyen and Yamada, 2013 . To enhance sparsity in the estimate of the generalized eigenvector, the proposed adaptive algorithm maximizes a certain non-convex criterion with $\ell _{1}$ penalty. A convergence analysis is also given for the proposed algorithm with decaying weight. Numerical experiments show that the proposed algorithm improves the subspace tracking performance in the situation where the covariance matrix pencil has sparse principal generalized eigenvector and is effective for recent sparsity-aware eigenvector analysis, e.g., sparse PCA.

Given an arbitrary square matrix $A$, we obtain the leading terms of the asymptotic expansions in the small, real parameter $\varepsilon$ of multiplicative perturbations $\widehat{A}(\varepsilon)=(I+\varepsilon\,B)A(I+\varepsilon\,C)$ of $A$ for arbitrary matrices $B$ and $C$. The analysis is separated into two rather different cases, depending on whether the unperturbed eigenvalue is zero or not. It is shown that in either case the leading exponents are obtained from the partial multiplicities of the eigenvalue of interest, and the leading coefficients generically involve only appropriately normalized left and right eigenvectors of $A$ associated with that eigenvalue, with no need of generalized eigenvectors. Similar results are obtained for multiplicative perturbation of singular values as well.

[EN] The relationship among eigenvalues of a given square matrix A and the rank one updated matrix A+vkq⁎, where vk is an eigenvector of A associated with the eigenvalue λk and q is an arbitrary vector, was described by Brauer in 1952. In this work we study the relations between the Jordan structures of A and A+vkq⁎. More precisely, we analyze the generalized eigenvectors of the updated matrix in terms of the generalized eigenvectors of A, as well as the Jordan chains of the updated matrix. Further, we obtain similar results when we use a generalized eigenvector of A instead of the eigenvector vk, Supported by the Spanish DGI grant MTM2013-43678-P.

The conditions of solvability and the structure of the generalized Green operator of a linear Noetherian matrix differential-algebraic boundary-value problem are found. The sufficient conditions for reducibility of the generalized matrix differential-algebraic equation to the traditional differential-algebraic equation with the unknown function in the form of a vector-column are obtained. In the solution of a generalized matrix differential-algebraic boundary-value problem, the original conditions of solvability and the structure of the general solution of a matrix Sylvester-type equation are used.

We set forth solvability conditions and construction of the generalized Green’s operator for a Noetherian matrix linear differential-algebraic boundary value problem. Sufficient conditions of reducibility of a generalized matrix differential-algebraic operator to a conventional differential-algebraic equation with an unknown column vector are established. To solve a matrix differential-algebraic boundary value problem, we employ a special solvability conditions and the construction of a general solution to a matrix Sylvester-type equation.

We first prove a matrix identity concerning the blocks of generalized Jordan blocks and then give applications to some invariants of matrices. As a consequence, we reprove the well known fact that for an eigenvalue λ , its algebraic multiplicity is greater than or equal to its geometric multiplicity.

Let be a real symmetric matrix having the zero/non-zero pattern off-diagonal entries described by a graph G. We focus in this article on the number of the connected components of some subgraphs of G associated with an eigenvector of . We generalize some known results so-called ‘discrete nodal domain theorems’ in the literature.

This paper investigates the connection between the Kannan-Lipton Orbit Problem and the polynomial invariant generator algorithm PILA based on eigenvectors computation. Namely, we reduce the problem of generating linear and polynomial certificates of non-reachability for the Orbit Problem for linear transformations with coefficients in \(\mathbb Q\) to the generalized eigenvector problem. Also, we prove the existence of such certificates for any transformation with integer coefficients, which is not the case with rational coefficients.

We present a new formulation of the hyperbolic singular value decomposition (HSVD) for an arbitrary complex (or real) matrix without hyperexchange matrices and redundant invariant parameters. In our formulation, we use only the concept of pseudo-unitary (or pseudo-orthogonal) matrices. We show that computing the HSVD in the general case is reduced to calculation of eigenvalues, eigenvectors, and generalized eigenvectors of some auxiliary matrices. The new formulation is more natural and useful for some applications. It naturally includes the ordinary singular value decomposition., Comment: 15 pages

In this paper, we study the analyticity of the semisimple eigenvalue and the corresponding eigenvector functions of general analytic matrix-valued functions which are very important in many engineering applications such as the optimum design of dynamical structures, model updating, damage detecting, quantum mechanics, diffraction grating theory, medical imaging, and social network theory. We establish some new sufficient conditions on existence of analytic eigenvalue and eigenvector functions corresponding to semisimple eigenvalues of general analytic matrix-valued functions, which relaxes the condition in [P. Lancaster, A. S. Markus, and F. Zhou, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 606--626]. We also present a numerical method to compute the derivatives of the eigenvalue and eigenvector functions. Numerical performance of the method is illustrated by some numerical examples.

The problem of assigning in part common left eigenvector assignment by the state feedback control of linear multi-input systems is considered in the paper. Based on linear matrix inequalities characterization, the exposition of the problem is formulated to handle a common left eigenvector for switched modes. Limited by preexisting degrees of freedom, which factually makes mutually contingent subsets of the closed-loop eigenvalue spectrum of each mode, the design conditions for in part common left eigenvector assignment of all modes are derived in the terms of linear matrix inequalities. The procedure provides a way available in linear switched control system design while the state-space control structures take standard forms. An example is included to assess the properties of the technique and its application.