Interpretation Of Eigenvalues Research Articles

PSYCHOMETRIC PROPERTIES OF ANXIETY ABOUT AGING SCALE AMONG MALAYSIAN YOUTHS: A CONFIRMATORY FACTOR ANALYSIS

The Anxiety about Aging Scale (AAS) is a questionnaire developed based on multidimensional aging anxiety to measure anxiety towards aging. However, the AAS constructs and items vary depending on the study population. This study aimed to explore the validity and reliability of AAS through evaluation of exploratory factor analysis (EFA) and confirmatory factor analysis (CFA) among youths in Malaysia. A cross-sectional was conducted among 1988 university and college students in Klang Valley, Malaysia, selected using stratified proportionate random sampling. EFA results suggested four factors solution based on the interpretation of Eigenvalues and scree plot with 59.11% variance extracted. Results of CFA supported the four-factor model of revised 17-item AAS with acceptable model fit indices and high factor loading. The revised 17-item AAS has good reliability through the assessment of the internal consistency of the items. In conclusion, the revised 17-item AAS measures four distinct factors is a valid and reliable questionnaire to measure anxiety towards aging among Malaysian youths

Theory of characteristic modes for lossy structures: Formulation and interpretation of eigenvalues

AbstractVolume integral equation (VIE) and surface integral equation (SIE) based characteristic mode (CM) formulations are investigated in the case of lossy objects. Imperfectly conducting metallic structures modelled with an impedance boundary condition and lossy dielectric bodies are considered. Two types of CM formulations are studied. In the first one, the generalized eigenvalue equation is expressed in terms of the Hermitian parts of the integral operators. In the second one, the weighting operator of the eigenvalue equation is defined so that the eigenvectors form a weighted orthogonal set, weighted with respect to radiated power. The first approach gives real eigensolutions and is found to lead to clustering of the eigenvalues as losses are increased. From these solutions it is difficult to separate contributions of radiated, reactive, and dissipated power. This separation appears naturally in the second approach that gives complex eigensolutions. As applications including lossy materials, CM analyses of a graphene sheet and a plasmonic nanoparticle are presented.

Fourier Expansion of the Beam Propagation Operator in the Eigenvalue Domain

Propagation methods in the frequency domain are powerful tools for the analysis of electromagnetic field propagation within photonic waveguide devices. They can be classified as wide angle beam propagation or eigenvector expansion methods. In comparison to the wide angle beam propagation methods, eigenvector expansion methods inherently include an accurate computation of radiation waves, guided modes, and evanescent waves. Common numerical methods rely on the solution of eigenvalue problems or the solution of matrix equations of high order. Consequently, the computation time needed can be tedious. A method is proposed, which exploits the inherent advantage of eigenvector expansion methods, but reduces the computation time considerably as the resulting propagation algorithm is based on simple matrix vector multiplications, allowing a large number of discretization points. The method takes advantage of the compact eigenvalue spectrum of the system matrix. The operator can be periodically expanded so that a Fourier decomposition can be applied. The expansion is carried out under consideration of the physical interpretation of eigenvalues for radiation waves, guided modes, and evanescent waves.

On Baxter $${\mathcal Q}$$ -Operators and their Arithmetic Implications

We consider Baxter \({\mathcal Q}\) -operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean L-functions proposed recently is generalized to the case of affine Lie algebras. We also introduce a simple generalization of Baxter operators and local L-functions compatible with this identification. This gives a connection of the Toda chain Baxter \({\mathcal Q}\) -operators with an Archimedean version of the Polya–Hilbert operator proposed by Berry-Keating. We also elucidate the Dorey–Tateo spectral interpretation of eigenvalues of \({\mathcal Q}\) -operators. Using explicit expressions for eigenfunctions of affine/relativistic Toda chain we obtain an Archimedean analog of Casselman–Shalika–Shintani formula for Whittaker function in terms of characters.

Rotations in Space and Orthogonal Matrices

One application of elementary linear algebra that never fails to surprise my students is that a composition of rotations in space is itself a rotation. The more general result that any 3 by 3 orthogonal matrix of determinant one (a proper orthogonal transformation) is a rotation can be demonstrated most convincingly if one has a computer package which will do the messy arithmetic involved. In this lesson we find the axis and angle of rotation of such a transformation. It brings together such different concepts from linear algebra as orthogonal bases and matrices, the geometric interpretation of eigenvalues and eigenvectors (both real and complex), and the geometric meaning of changing bases. We presume that the students have seen the 2 by 2 matrix of a rotation in the plane and that they know a little about orthogonal matrices and eigenvalues and eigenvectors.

Notes on the Use and Interpretation of Discriminant Analysis

This article focuses on two problems in applications of discriminant analysis in political research: first, evaluating the success of classification, and, second, determining the importance of predictor variables when the classification variable is a polychotomy. Discussion of the former rejects the existing procedures for evaluating classification success and advances an empirical simulation technique that is more appropriate. Discussion of the latter focuses on canonical correlation analysis and the misleading and inappropriate conclusions that may be drawn when the categorical variable is a polychotomy. Most important, this second discussion advances an alternative interpretation of eigenvalues; presents the redundancy measure, which is the appropriate measure of association between the independent variable set and the multi-categorical variable set; and suggests a method of evaluating the importance of the independent variables and the multi-categories of the group variable. We illustrate our discussion with random number data. Finally, we apply the techniques advanced in a public policy analysis.

Scaling variables and interpretation of eigenvalues in principal component analysis of geologic data

The dominant feature distinguishing one method of principal components analysis from another is the manner in which the original data are transformed prior to the other computations. The only other distinguishing feature of any importance is whether the eigenvectors of the inner product-moment of the transformed data matrix are taken directly as the Q-mode scores or scaled by the square roots of their associated eigenvalues and called the R-mode loadings. If the eigenvectors are extracted from the product-moment correlation matrix, the variables, in effect, were transformed by column standardization (zero means and unit variances), and the sum of the p-largest eigenvalues divided by the sum of all the eigenvalues indicates the degree to which a model containing pcomponents will account for the total variance in the original data. However, if the data were transformed in any manner other than column standardization, the eigenvalues cannot be used in this manner, but can only be used to determine the degree to which the model will account for the transformed data. Regardless of the type of principal components analysis that is performed—even whether it is Ror Q-mode—the goodness-of-fit of the model to the original data is given better by the eigenvalues of the correlation matrix than by those of the matrix that was actually factored.