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Matrix with Prescribed Eigenvectors. It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…. Covariance expressions for eigenvalue and eigenvector problems.
There are a number of important scientific and engineering problems whose solutions take the form of an eigenvalue-- eigenvector problem. Some notable examples include solutions to linear systems of ordinary differential equations, controllability of linear systems, finite element analysis, chemical kinetics, fitting ellipses to noisy data, and optimal estimation of attitude from unit vectors. In many of these problems, having knowledge of the eigenvalue and eigenvector Jacobians is either necessary or is nearly as important as having the solution itself.
For instance, Jacobians are necessary to find the uncertainty in a computed eigenvalue or eigenvector estimate. This uncertainty, which is usually represented as a covariance matrix, has been well studied for problems similar to the eigenvalue and eigenvector problem, such as singular value decomposition. There has been substantially less research on the covariance of an optimal estimate originating from an eigenvalue- eigenvector problem.
In this thesis we develop two general expressions for the Jacobians of eigenvalues and eigenvectors with respect to the elements of their parent matrix. The expressions developed make use of only the parent matrix and the eigenvalue and eigenvector pair under consideration. In addition, they are applicable to any general matrix including complex valued matrices, eigenvalues, and eigenvectors as long as the eigenvalues are simple.
Alongside this, we develop expressions that determine the uncertainty in a vector estimate obtained from an eigenvalue- eigenvector problem given the uncertainty of the terms of the matrix.
The Jacobian expressions developed are numerically validated with forward finite, differencing and the covariance expressions are validated using Monte Carlo analysis. Finally, the results from this work are used to determine covariance expressions for a variety of estimation problem examples and are also applied to the design of a dynamical system.
Motivating the Concept of Eigenvectors via Cryptography. New methods of teaching linear algebra in the undergraduate curriculum have attracted much interest lately. Most of this work is focused on evaluating and discussing the integration of special computer software into the Linear Algebra curriculum.
In this article, I discuss my approach on introducing the concept of eigenvectors and eigenvalues,…. Eigenvectors phase correction in inverse modal problem.
The solution of the inverse modal problem for the spatial parameters of mechanical and structural systems is heavily dependent on the quality of the modal parameters obtained from the experiments. While experimental and environmental noises will always exist during modal testing, the resulting modal parameters are expected to be corrupted with different levels of noise.
A novel methodology is presented in this work to mitigate the errors in the eigenvectors when solving the inverse modal problem for the spatial parameters.
The phases of the eigenvector component were utilized as design variables within an optimization problem that minimizes the difference between the calculated and experimental transfer functions. The equation of motion in terms of the modal and spatial parameters was used as a constraint in the optimization problem.
Constraints that reserve the positive and semi-positive definiteness and the inter-connectivity of the spatial matrices were implemented using semi-definite programming. The results showed that the proposed method is superior when compared with a known method in the literature.
Eigenvector space model to capture features of documents. Directory of Open Access Journals Sweden. Full Text Available Eigenvectors are a special set of vectors associated with a linear system of equations. Because of the special property of eigenvector , it has been used a lot for computer vision area.
When the eigenvector is applied to information retrieval field, it is possible to obtain properties of documents data corpus. To capture properties of given documents, this paper conducted simple experiments to prove the eigenvector is also possible to use in document analysis. For the experiment, we use short abstract document of Wikipedia provided by DBpedia as a document corpus. To build an original square matrix, the most popular method named tf-idf measurement will be used.
After calculating the eigenvectors of original matrix, each vector will be plotted into 3D graph to find what the eigenvector means in document processing. Distinct types of eigenvector localization in networks. The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks.
Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. Full Text Available Graph-based subspace learning is a class of dimensionality reduction technique in face recognition. The technique reveals the local manifold structure of face data that hidden in the image space via a linear projection. However, the real world face data may be too complex to measure due to both external imaging noises and the intra-class variations of the face images.
Hence, features which are extracted by the graph-based technique could be noisy. An appropriate weight should be imposed to the data features for better data discrimination. In this paper, a piecewise weighting function, known as Eigenvector Weighting Function EWF, is proposed and implemented in two graph based subspace learning techniques, namely Locality Preserving Projection and Neighbourhood Preserving Embedding.
Specifically, the computed projection subspace of the learning approach is decomposed into three partitions: a subspace due to intra-class variations, an intrinsic face subspace, and a subspace which is attributed to imaging noises. Projected data features are weighted differently in these subspaces to emphasize the intrinsic face subspace while penalizing the other two subspaces.
Localized eigenvectors of the non-backtracking matrix. In the case of graph partitioning, the emergence of localized eigenvectors can cause the standard spectral method to fail. To overcome this problem, the spectral method using a non-backtracking matrix was proposed. Based on numerical experiments on several examples of real networks, it is clear that the non-backtracking matrix does not exhibit localization of eigenvectors. However, we show that localized eigenvectors of the non-backtracking matrix can exist outside the spectral band, which may lead to deterioration in the performance of graph partitioning.
Use of eigenvectors in understanding and correcting storage ring orbits. Since A is not necessarily a symmetric or even a square matrix we symmetrize it by using A T A. Then we find the eigenvalues and eigenvectors of this A T A matrix. The physical interpretation of the eigenvectors for circular machines is discussed. We are presenting a method, in which the kick vector is expressed as linear combination of the eigenvectors. An additional advantage of this method is that it yields the smallest possible kick vector to correct the orbit.
It will be evident, that the accuracy of this method allows the combination of the global orbit correction and local optimization of the orbit for beam lines and insertion devices. The eigenvector decomposition can also be used for optimizing kick vectors, taking advantage of the fact that eigenvectors with corresponding small eigenvalue generate negligible orbit changes.
Thus, one can reduce a kick vector calculated by any other correction method and still stay within the tolerance for orbit correction.
The use of eigenvectors in accurately measuring the response matrix and the use of the eigenvalue decomposition orbit correction algorithm in digital feedback is discussed.
Image denoising via adaptive eigenvectors of graph Laplacian. An image denoising method via adaptive eigenvectors of graph Laplacian EGL is proposed. Unlike the trivial parameter setting of the used eigenvectors in the traditional EGL method, in our method, the eigenvectors are adaptively selected in the whole denoising procedure. In detail, a rough image is first built with the eigenvectors from the noisy image, where the eigenvectors are selected by using the deviation estimation of the clean image.
Subsequently, a guided image is effectively restored with a weighted average of the noisy and rough images. In this operation, the average coefficient is adaptively obtained to set the deviation of the guided image to approximately that of the clean image. Finally, the denoised image is achieved by a group-sparse model with the pattern from the guided image, where the eigenvectors are chosen in the error control of the noise deviation.
Moreover, a modified group orthogonal matching pursuit algorithm is developed to efficiently solve the above group sparse model. The experiments show that our method not only improves the practicality of the EGL methods with the dependence reduction of the parameter setting, but also can outperform some well-developed denoising methods, especially for noise with large deviations. The package can determine the Eigen-system of complex general, complex Hermitian, real general, real symmetric, real symmetric band, real symmetric tridiagonal, special real tridiagonal, generalized real, and generalized real symmetric matrices.
In addition, there are two routines which use the singular value decomposition to solve certain least squares problem. A teaching proposal for the study of Eigenvectors and Eigenvalues. Full Text Available In this work, we present a teaching proposal which emphasizes on visualization and physical applications in the study of eigenvectors and eigenvalues. These concepts are introduced using the notion of the moment of inertia of a rigid body and the GeoGebra software.
It was designed following a particular sequence of activities with the schema: exploration, introduction of concepts, structuring of knowledge and application, and considering the three worlds of mathematical thinking provided by Tall: embodied, symbolic and formal.
Full Text Available In this contribution we give an explicit formula for the eigenvectors of Hamiltonians of open Bazhanov-Stroganov quantum chain. We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigen-spaces of a symmetric positive definite matrix.
In our applications, the dimension of a matrix is large and the cost of its inverting is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning.
Estimates will be provided which show that the preconditioned method converges linearly and uniformly in the matrix dimension when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.
We obtain renormalized sets of right and left eigenvectors of the flux vector Jacobians of the relativistic MHD equations, which are regular and span a complete basis in any physical state including degenerate ones. The renormalization procedure relies on the characterization of the degeneracy types in terms of the normal and tangential components of the magnetic field to the wave front in the fluid rest frame.
Proper expressions of the renormalized eigenvectors in conserved variables are obtained through the corresponding matrix transformations. Our work completes previous analysis that present different sets of right eigenvectors for non-degenerate and degenerate states, and can be seen as a relativistic generalization of earlier work performed in classical MHD.
Based on the full wave decomposition FWD provided by the renormalized set of eigenvectors in conserved variables, we have also developed a linearized Roe-type Riemann solver. Extensive testing against one- and two-dimensional standard numerical problems allows us to conclude that our solver is very robust. When compared with a family of simpler solvers that avoid the knowledge of the full characteristic structure of the equations in the computation of the numerical fluxes, our solver turns out to be less diffusive than HLL and HLLC, and comparable in accuracy to the HLLD solver.
The amount of operations needed by the FWD solver makes it less efficient computationally than those of the HLL family in one-dimensional problems. However, its relative efficiency increases in multidimensional simulations.
The best of both worlds: Phylogenetic eigenvector regression and mapping. Full Text Available Eigenfunction analyses have been widely used to model patterns of autocorrelation in time, space and phylogeny. In a phylogenetic context, Diniz-Filho et al. More recently, a new approach called Phylogenetic Eigenvector Mapping PEM was proposed, with the main advantage of explicitly incorporating a model-based warping in phylogenetic distance in which an Ornstein-Uhlenbeck O-U process is fitted to data before eigenvector extraction.
Here we compared PVR and PEM in respect to estimated phylogenetic signal, correlated evolution under alternative evolutionary models and phylogenetic imputation, using simulated data.
Despite similarity between the two approaches, PEM has a slightly higher prediction ability and is more general than the original PVR.
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