Search

We propose a new symmetric variational functional-algebraic statement of the eigenvalue problem in a Hilbert space with a linear dependence on the spectral parameter for a class of mathematical models of thin-walled structures with an attached oscillator. The existence of eigenvalues and eigenvectors is established. A new symmetric approximation of the problem in a finite-dimensional subspace with a linear dependence on the spectral parameter is constructed. Error estimates are obtained for the approximate eigenvalues and eigenvectors. The theoretical results are illustrated with an example of a structural mechanics problem. [ABSTRACT FROM AUTHOR]

As an extension of Sylvester's matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form. [ABSTRACT FROM AUTHOR]

The spectral decomposition of a real skew-symmetric matrix is shown to be equivalent to a specific structured singular value decomposition (SVD) of the matrix. Based on such equivalence, we propose a skew-symmetric Lanczos bidiagonalization (SSLBD) method to compute extremal singular values and the corresponding singular vectors of the matrix, from which its extremal conjugate eigenpairs are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of the computed left and right Lanczos basis vectors and the semi-biorthogonality of two sets of basis vectors are needed to compute the singular values accurately and to make the method work as if it does in exact arithmetic. A commonly used efficient partial reorthogonalization strategy is adapted to maintain the desired semi-orthogonality and semi-biorthogonality. For practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a Toeplitz construction method based on eigenvalues and eigenvectors is proposed to combine with traditional denoising algorithms, including fractional low-order moment (FLOM), phased fractional low-order moment (PFLOM), and correntropy-based correlation (CRCO) methods. It can improve the direction of arrival (DOA) estimation of signals in impulsive noise. Firstly, the algorithm performs eigenvalue decomposition on the received covariance matrix to obtain eigenvectors and eigenvalues, and then the Toeplitz matrix is created according to the eigenvectors corresponding to its eigenvalues. Secondly, the spatial averaging method is used to obtain an unbiased estimate of the Toeplitz matrix, which is then weighted and added based on the corresponding eigenvalues. Next, the noise subspace of the Toeplitz matrix is reconstructed to obtain the one that has less angle information. Finally, the DOA of the coherent signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. The improved method based on the Toeplitz matrix can not only suppress the effect of impulsive noise but can also solve the problem of aperture loss due to its decoherence. A series of simulations have shown that they have better performances than other algorithms. [ABSTRACT FROM AUTHOR]

In this paper we expressed the eigenvalues of a sort of heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From the prescribed eigenvalues, we computed eigenvectors for these types of matrices, giving also a formula not dependent on any unknown parameter for its determinant and inverse. Potential applications of the results are still provided.

Let R_ε=R∪{-∞}, with R being a set of all real numbers. The algebraic structure (R_ε,⊕,⊗) is called max-plus algebra. The task of finding the eigenvalue and eigenvector is called the eigenproblem. There are several methods developed to solve the eigenproblem of A∈R_ε^(n×n), one of them is by using the characteristic max-polynomial. There is another algebraic structure that is isomorphic with max-plus algebra, namely min-plus algebra. Min-plus algebra is a set of R_(ε^' )=R∪{+∞} that uses minimum (⊕^' ) and addition (⊗) operations. The eigenproblem in min-plus algebra is to determine λ∈R_(ε^' ) and v∈R_(ε^')^n such that A⊗v=λ⊗v. In this paper, we provide a method for determining the characteristic min-polynomial and solving the eigenproblem by using the characteristic min-polynomial. We show that the characteristic min-polynomial of A∈R_(ε^')^(n×n) is the permanent of I⊗x⊕^' A, the smallest corner of χ_A (x) is the principal eigenvalue (λ(A)), and the columns of A_λ^+ with zero diagonal elements are eigenvectors corresponding to the principal eigenvalue.

The eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first necessary to be sure what conditions must be satisfied, so a short review of density operators is given. Since position and momentum do not commute in quantum mechanics, it is useful to start with the consequences of the noncommutation, which is generally the Schrödinger-Robertson uncertainty relation, which is also briefly reviewed. It is found that the question of whether a gaussian operator is a density operator is directly tied to this uncertainty relation. Since the Wigner function is the phase space representation of a translational density operator, it is natural to consider the gaussian phase space function associated with a gaussian operator and to compare the phase space and operator properties. Throughout such discussions, the independent parameters in these functions are the first and second moments of position and momentum. The application of this formalism to the free translation and spreading of a gaussian packet is given and shows the formal similarity between classical and quantum behavior, whereas the literature standardly only considers the pure state case (equivalent to a single wavefunction).

In order to guarantee the downloading quality requirements of users and improve the stability of data transmission in a BitTorrent-like peer-to-peer file sharing system, this article deals with eigenproblems of addition-min algebras. First, it provides a sufficient and necessary condition for a vector being an eigenvector of a given matrix, and then presents an algorithm for finding all the eigenvalues and eigenvectors of a given matrix. It further proposes a sufficient and necessary condition for a vector being a constrained eigenvector of a given matrix and supplies an algorithm for computing all the constrained eigenvectors and eigenvalues of a given matrix. This article finally discusses the supereigenproblem of a given matrix and presents an algorithm for obtaining the maximum constrained supereigenvalue and depicting the feasible region of all the constrained supereigenvectors for a given matrix. It also gives some examples for illustrating the algorithms, respectively. [ABSTRACT FROM AUTHOR]

Let G be a connected graph. Denote by d i the degree of a vertex v i in G. Let f (x , y) > 0 be a real symmetric function. Consider an edge-weighted graph in such a way that for each edge v i v j of G, the weight of v i v j is equal to the value f (d i , d j) . Therefore, we have a degree-based weighted adjacency matrix A f (G) of G, in which the (i, j)-entry is equal to f (d i , d j) if v i v j is an edge of G and is equal to zero otherwise. Let x be a positive eigenvector corresponding to the largest eigenvalue λ 1 (A f (G)) of the weighted adjacency matrix A f (G) . In this paper, we first consider the unimodality of the eigenvector x on an induced path of G. Second, if f(x, y) is increasing in the variable x, then we investigate how the largest weighted adjacency eigenvalue λ 1 (A f (G)) changes when G is perturbed by vertex contraction or edge subdivision. The aim of this paper is to unify the study of spectral properties for the degree-based weighted adjacency matrices of graphs. [ABSTRACT FROM AUTHOR]

. We consider a minimal realization of a rational matrix. We perturb all the coefficients of matrix polynomial and some coefficients from the realization part present in the realization form of rational matrix. We derive explicit computable formulae for backward error of approximate eigenvalues and eigenpairs of regular rational matrix. We also determine minimal perturbations for all the coefficients of matrix polynomial and some coefficients from the realization part for which approximate eigenvalues are exact eigenvalues of the perturbed rational matrix. [ABSTRACT FROM AUTHOR]

This paper has proposed a comprehensive indicator based on principal component analysis (PCA) for diagnosing the state of anaerobic digestion. Various state and performance variables were monitored under different operational modes, including start-up, interruption and resumption of substrate supply, and impulse organic loading rates. While these individual variables are useful for estimating the state of anaerobic digestion, they must be interpreted by experts. Coupled indicators combine these variables with the effect of offering more detailed insights, but they are limited in their universal applicability. Time-series eigenvalues reflected the anaerobic digestion process occurring in response to operational changes: Stable states were identified by eigenvalue peaks below 1.0, and they had an average below 0.2. Slightly perturbed states were identified by a consistent decrease in eigenvalue peaks from a value of below 4.0 or by observing isolated peaks below 3.0. Disturbed states were identified by repeated eigenvalue peaks over 3.0, and they had an average above 0.6. The long-term persistence of these peaks signals an increasing kinetic imbalance, which could lead to process failure. Ultimately, this study demonstrates that time-series eigenvalue analysis is an effective comprehensive indicator for identifying kinetic imbalances in anaerobic digestion. [ABSTRACT FROM AUTHOR]

In order to guarantee the downloading quality requirements of users and improve the stability of data transmission in a BitTorrent-like peer-to-peer file sharing system, this article deals with eigenproblems of addition-min algebras. First, it provides a sufficient and necessary condition for a vector being an eigenvector of a given matrix, and then presents an algorithm for finding all the eigenvalues and eigenvectors of a given matrix. It further proposes a sufficient and necessary condition for a vector being a constrained eigenvector of a given matrix and supplies an algorithm for computing all the constrained eigenvectors and eigenvalues of a given matrix. This article finally discusses the supereigenproblem of a given matrix and presents an algorithm for obtaining the maximum constrained supereigenvalue and depicting the feasible region of all the constrained supereigenvectors for a given matrix. It also gives some examples for illustrating the algorithms, respectively. [ABSTRACT FROM AUTHOR]

In this paper we expressed the eigenvalues of a sort of heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From the prescribed eigenvalues, we computed eigenvectors for these types of matrices, giving also a formula not dependent on any unknown parameter for its determinant and inverse. Potential applications of the results are still provided. [ABSTRACT FROM AUTHOR]

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel's, Helmholtz's, Laplace's, Legendre's, Poisson's, and Schrödinger's equations. The algorithm extracting the first principal component is also provided. [ABSTRACT FROM AUTHOR]

A new symmetric linear variational statement is proposed for the problem of eigenvibrations of a plate with an attached oscillator. The existence of the sequence of positive eigenvalues of finite multiplicity with the limit point at infinity and the corresponding complete orthonormal system of eigenvectors is established. A new symmetric scheme of the finite element method with Hermite finite elements is stated. Error estimates consistent with the solution smoothness for the approximate eigenvalues and approximate eigenvectors are proved. The results of numerical experiments illustrating the influence of the smoothness of the solution on the computation accuracy are presented. [ABSTRACT FROM AUTHOR]

For the problem of eigenvibrations of the shallow shell with an attached oscillator, a new symmetric variational statement in the Hilbert space was proposed. It was established that there exist a sequence of positive eigenvalues of finite multiplicity with a limit point at infinity and the corresponding complete orthonormal system of eigenvectors. The problem was approximated by the mesh scheme of the finite element method with Hermite finite elements. Theoretical error estimates for the approximate solutions were proved. The theoretical findings were verified by the results of numerical experiments.

The purpose of this article was to introduce the basic concepts, calculation methods, two examples and SAS implementation related to the principal component analysis. Basic concepts included correlation matrix, eigenvalues and eigenvectors, principal component variables, principal component expressions and principal component properties. The calculation method involved the calculation of eigenvalues and eigenvectors, the calculation principle and the coefficient estimation and the number determination of the principal component. The data in the two examples were measurement results of 4 liver function indicators in 20 patients with liver disease and survey results of literature metrology indicators in 23 tumor journals. With the help of SAS software, the principal component analysis was carried out on the quantitative data in the two cases, and based on the calculation results of the principal components, the sample clustering and sample sorting were respectively realized, and a reasonable explanation was given for the output results. [ABSTRACT FROM AUTHOR]

The eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first necessary to be sure what conditions must be satisfied, so a short review of density operators is given. Since position and momentum do not commute in quantum mechanics, it is useful to start with the consequences of the noncommutation, which is generally the Schrödinger-Robertson uncertainty relation, which is also briefly reviewed. It is found that the question of whether a gaussian operator is a density operator is directly tied to this uncertainty relation. Since the Wigner function is the phase space representation of a translational density operator, it is natural to consider the gaussian phase space function associated with a gaussian operator and to compare the phase space and operator properties. Throughout such discussions, the independent parameters in these functions are the first and second moments of position and momentum. The application of this formalism to the free translation and spreading of a gaussian packet is given and shows the formal similarity between classical and quantum behavior, whereas the literature standardly only considers the pure state case (equivalent to a single wavefunction). [ABSTRACT FROM AUTHOR]

The seismic coherence attribute is one of the most typically used seismic discontinuity feature detection technologies, which is widely used in fault detection, channel boundary characterization, and other occasions. The eigen-structure-based coherence algorithm possesses the property of stability by using the eigenvalues of seismic data’s covariance matrix, but in the algorithm, only the eigenvalues are used to calculate the final coherence value, and the information contained in eigenvectors is ignored. Through analysis, it is found that the elements in the eigenvectors represent the energy differences of seismic traces. By directly measuring the difference of different elements in an eigenvector, the energy differences between different seismic records can be effectively measured. Since the seismic data on both sides of some discontinuous boundaries, such as channel boundaries, principally show amplitude differences, the coherence properties based on eigenvectors can better characterize them. Through theoretical analysis, model testing, and case application, this letter illustrates the reliability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This note is intended as a contribution to the study of quantitative measures of graph complexity that use entropy measures based on symmetry. Determining orbit sizes of graph automorphism groups is a key part of such studies. Here we focus on an extreme case where every orbit contains just a single vertex. A permutation of the vertices of a graph G is an automorphism if, and only if, the corresponding permutation matrix commutes with the adjacency matrix of G. This fact establishes a direct connection between the adjacency matrix and the automorphism group. In particular, it is known that if the eigenvalues of the adjacency matrix of G are all distinct, every non-trivial automorphism has order 2. In this note, we add a condition to the case of distinct eigenvalues that makes the graph asymmetric, i.e., reduces the automorphism group to the identity alone. In addition, we prove analogous results for the Google and Laplacian matrices. The condition is used to build an O(n3) algorithm for detecting identity graphs, and examples are given to demonstrate that it is sufficient, but not necessary. [ABSTRACT FROM AUTHOR]

The paper is the first to use principal component analysis (PCA) in veterinary epidemiology and aims to identify the most significant agents of diarrhoea in calves. Data were collected from 245 calves on 32 farms within 13 districts of Northern Kazakhstan. Epidemiological data were simulated in R, using standard PCA modules. The data clustering was carried out based on the prevalence of seven enteropathogens for an age group dataset (represented by four animal groups) and a farm type dataset (by three farm types depending on herd size). The simulation revealed that the components identified here for one and two-week calves explained 91.31% of the variance. The first two components of large and middle-sized farms covered 90.3% of the variance. The coordinates corresponding to pathogens were approximately visualised in directions of eigenvectors for each age group and farm type. The coordinate for Cryptosporidium parvum was in directions of eigenvectors for one-to three-week calves and large farms. The coordinate for rotaviruses was in the direction of eigenvectors for four-week calves and medium-sized and small farms. So, PCA can potentially be useful in the clustering of epidemiological datasets and making decisions on control of infectious diseases with a multi-pathogenic nature. [ABSTRACT FROM AUTHOR]

In agricultural experiments, if outliers are present in a data set, inference of experiment may be reversed. The purpose of this article is to develop a method for detection of subset of outlier vectors in block designs for incomplete multiresponse experiments in presence of masking. We defined an influence matrix comprising of Cook-statistics in its diagonal and product of two Cook-statistics in its off-diagonal positions. We then obtained eigenvectors corresponding to large eigenvalues of this matrix which can be used for identification of influential subsets. The proposed procedure has been illustrated with a real life data set. [ABSTRACT FROM AUTHOR]

In this paper, a Toeplitz construction method based on eigenvalues and eigenvectors is proposed to combine with traditional denoising algorithms, including fractional low-order moment (FLOM), phased fractional low-order moment (PFLOM), and correntropy-based correlation (CRCO) methods. It can improve the direction of arrival (DOA) estimation of signals in impulsive noise. Firstly, the algorithm performs eigenvalue decomposition on the received covariance matrix to obtain eigenvectors and eigenvalues, and then the Toeplitz matrix is created according to the eigenvectors corresponding to its eigenvalues. Secondly, the spatial averaging method is used to obtain an unbiased estimate of the Toeplitz matrix, which is then weighted and added based on the corresponding eigenvalues. Next, the noise subspace of the Toeplitz matrix is reconstructed to obtain the one that has less angle information. Finally, the DOA of the coherent signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. The improved method based on the Toeplitz matrix can not only suppress the effect of impulsive noise but can also solve the problem of aperture loss due to its decoherence. A series of simulations have shown that they have better performances than other algorithms.

In this article, we express the eigenvalues of real antitridiagonal Hankel matrices as the zeros of given rational functions. We still derive eigenvectors for these structured matrices at the expense of prescribed eigenvalues.

Solving the eigenvalue problem is particularly important in almost all fields of science and engineering. With the development of quantum computers, multiple algorithms have been proposed for this purpose. However, such methods are usually only applicable to matrices of specific types, such as unitary or Hermitian matrices. The quantum annealer of the D-Wave, a quantum computer, returns the minimum value of the quadratic unconstrained binary optimization (QUBO) model. Thus, quantum annealers can be leveraged to solve arbitrary eigenvalue problems by formulating corresponding QUBO models. In this paper, we propose two higher-order unconstrained optimization (HUBO) formulations to solve eigenvalue problems involving n×ngeneral matrices. In addition, we use a formula to reduce the order and convert the HUBO model into a QUBO model. Further, by using a quantum approximate optimization algorithm, this method can be extended to a gate-model quantum computer.

Distributed energy resource (DER) has become an effective attempt in promoting use of renewable energy resources for electricity generation. The core intention of this study is to expand an approach for optimally placing several DER units to attain the most stable performance of the system and the greatest power losses decrease. The recommended technique is established on two analytical methods for analyzing voltage stability: the new modified modal analysis (MMA) and the continuation power flow (CPF) or MMA–CPF methods. The MMA evaluates voltage stability by considering incremental connection relating voltage and active power, which includes the eigenvalue and the related eigenvectors computed from the reduced modified Jacobian matrix. Furthermore, an active participation factor (APF) is computed from the eigenvectors of the reduced modified Jacobian matrix. The CPF method uses a predictor–corrector stepping pattern to reach the solution track and compute the tangent vector sensitivity (TVS). Both APF and TVS indicate each load bus sensitivity in the network. In addition, an objective function regarding losses decrease and eigenvalue is expressed to calculate the best bus position for DER allocation. The proposed MMA–CPF technique has been assessed on a 34-bus RDN and the outcomes demonstrate the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We investigate the shape and strength of the magnetic fabrics (anisotropy of magnetic susceptibility (AMS) data) of various massive granitic plutons from different parts of India, using the eigenvalue method. The study aims to analyse eigenvalues and establish their relationship with various deformational attributes. It involves: (1) calculating eigenvectors and their corresponding eigenvalues from magnetic fabric datasets; (2) finding a link between the geometrical appearance of eigenvectors and the mechanistic issues involved with a specific deformation scenario; and (3) determining shape and strength parameters from the magnetic foliation data distribution. The statistical analysis for the unimodal magnetic fabric dataset of orthorhombic symmetry class implies that the plane, consisting of intermediate (V2) and minimum (V3) eigenvectors with pole V1, accurately traces the instantaneous stretching axis (ISAmax) of a particular material flow system under a pure shear regime. Moreover, for the distributions of similar symmetry and modality, we infer that the rotational characteristics of eigenvectors with respect to a fixed coordinate cause a distinct shift of such planes (V2–V3) from the ISAmax of a steady-state flow system under simple shear, where a substantial amount of rotational strain is involved. However, our findings also suggest that variation in symmetry and modality of magnetic fabric data distribution of different studied granitoids can directly influence the relative disposition of V2–V3 with respect to the direction of ISAmax. We conclude that eigenvalue analysis of magnetic fabrics is a powerful approach, which can be utilized while studying the salient deformational aspects of any syntectonic massive granitic body. [ABSTRACT FROM AUTHOR]

In this paper, we present some results on fixed point index calculations for multivalued mappings and apply them to prove the existence of solutions to multivalued equations in ordered space, under flexible conditions for the positive eigenvalue.

The notion of root polynomials of a polynomial matrix P (λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R (λ) , possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R (λ) , without any need to assume that R (λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R (λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ 0. If λ 0 is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations. [ABSTRACT FROM AUTHOR]

Various trace minimization principles have interplayed with numerical computations for the standard eigenvalue and generalized eigenvalue problems in general, as well as important applied eigenvalue problems including the linear response eigenvalue problem from electronic structure calculation and the symplectic eigenvalue problem of positive definite matrices that play important roles in classical Hamiltonian dynamics, quantum mechanics, and quantum information, among others. In this paper, Ky Fan's trace minimization principle is extended along the line of the Brockett cost function tr (D X H A X) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we investigate inf X ⁡ tr (D X H A X) subject to X H B X = I k (the k × k identity matrix) or X H B X = J k , where J k = diag (± 1). We establish conditions under which the infimum is finite and when it is finite, analytic solutions are obtained in terms of the eigenvalues and eigenvectors of the matrix pencil A − λ B , where B is possibly indefinite and possibly singular, and D is also possibly indefinite. [ABSTRACT FROM AUTHOR]

In crystallography, piezoelectric tensors of various crystals play a crucial role in piezoelectric effect and converse piezoelectric effect. Generally, a third order real tensor is called a piezoelectric-type tensor if it is partially symmetric with respect to its last two indices. The piezoelectric tensor is a piezoelectric-type tensor of dimension three. We introduce C-eigenvalues and C-eigenvectors for piezoelectric-type tensors. Here, "C'' names after Curie brothers, who first discovered the piezoelectric effect. We show that C-eigenvalues always exist, they are invariant under orthogonal transformations, and for a piezoelectric-type tensor, the largest C-eigenvalue and its C-eigenvectors form the best rank-one piezoelectric-type approximation of that tensor. This means that for the piezoelectric tensor, its largest C-eigenvalue determines the highest piezoelectric coupling constant. We further show that for the piezoelectric tensor, the largest C-eigenvalue corresponds to the electric displacement vector with the largest 2-norm in the piezoelectric effect under unit uniaxial stress, and the strain tensor with the largest 2-norm in the converse piezoelectric effect under unit electric field vector. Thus, C-eigenvalues and C-eigenvectors have concrete physical meanings in piezoelectric effect and converse piezoelectric effect. Finally, by numerical experiments, we report C-eigenvalues and associated C-eigenvectors for piezoelectric tensors corresponding to several piezoelectric crystals. [ABSTRACT FROM AUTHOR]

In the literature it is common to use the first and last pencils D 1 (λ , P) and D k (λ , P) in the “standard basis” for the vector space D L (P) of block-symmetric pencils to solve the symmetric/Hermitian polynomial eigenvalue problem P (λ) x = 0 . When the polynomial P (λ) has odd degree, it was proven in recent years that the use of an alternative linearization T P is more convenient because it has better numerical properties and its use is more universal since T P is a strong linearization of any matrix polynomial P (λ) , while D 1 (λ ; P) and D k (λ ; P) are not. However, T P is not defined for even degree matrix polynomials. In this paper we consider the case when P (λ) has even degree. It is believed that the eigenpair backward errors for the linearization D 1 (λ ; P) and D k (λ ; P) cannot differ much from the backward error of the original problem. We show that this is not the case, even when the polynomial P (λ) is well-scaled because of the ill-conditioning of the eigenvectors of D 1 (λ ; P) and D k (λ ; P) . We introduce two block-symmetric linearizations for even degree matrix polynomials that overcome this problem and become an appropriate alternative to the traditional use of D 1 (λ ; P) and D k (λ ; P) . [ABSTRACT FROM AUTHOR]