Eigenvectors description

So fiir, we have described only situations with two classes. The method can also be applied to K classes. It is then sometimes called descriptive linear discriminant analysis. In this case the weight vectors can be shown to be the eigenvectors of the matrix ... 

An instructive description of the first-order perturbation treatment of the quadrupole interaction in Ni has been given by Travis and Spijkerman [3]. These authors also show in graphical form the quadrupole-spectrum line positions and the quadrupole-spectrum as a function of the asymmetry parameter r/ they give eigenvector coefficients and show the orientation dependence of the quadrupole-spectrum line intensities for a single crystal of a Ni compound. The reader is also referred to the article by Dunlap [15] about electric quadrupole interaction, in general. 

The quantities U Ix)j and UFx)j in (13) are projections of the eigenvector j along lx- From the above equations, we can interpret these as follows. The term UFx)j is the amount that the transition j received from the total X magnetization, created from the equilibrium state, and (U Ix)j is how much that transition contributes to the observed signal. These two terms may not be equal, as we see in exchanging systems. This general approach forms the basis of the description of dynamic NMR lineshapes. 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

The matrix Q can now be transformed into a stochastic matrix, which will be descriptive of the restricted random walks rather than of their generation employing probabilities based on unrestricted walk models. The transformation is performed as follows Let Xt be the largest eigenvalue of the matrix Q, and let Sj be the corresponding left-hand side eigenvector (defined by SjQ = X ). Let A be a diagonal matrix with elements a(i,j) = (/) 8st = [ 1(1),. v,(2),..., (v)] and 8(i,j) is the... 

Information about the symmetry of each energy level e, is contained in the eigenvector m, of the diagonalization matrix U. In a usual description, the transformation from the basis set of the atomic terms (AT) into the basis set... 

In the present paper we assume that the molecule has the icosahedral symmetry. If one wants to consider a distortion of C 0+ or Cb0. the energy levels and their eigenvectors obtained here can be used as a starting point for the description of the Jahn-Teller effect in these systems. Indeed, the electron-phonon (or vibronic) coupling occurs if [.Tei]2 contains Fvib [19]. (Here Fd is the symmetry of an electronic molecular term, while J b is the symmetry of a vibrational normal mode.) Calculations using the terms in scheme of Ref. [4] have been performed in Ref. [20]. 

Interatomic Force Constants (IFCs) are the proportionality coefficients between the displacements of atoms from their equilibrium positions and the forces they induce on other atoms (or themselves). Their knowledge allows to build vibrational eigenfrequencies and eigenvectors of solids. This paper describes IFCs for different solids (SiC>2-quartz, SiC>2-stishovite, BaTiC>3, Si) obtained within the Local-Density Approximation to Density-Functional Theory. An efficient variation-perturbation approach has been used to extract the linear response of wavefunctions and density to atomic displacements. In mixed ionic-covalent solids, like SiC>2 or BaTiC>3, the careful treatment of the long-range IFCs is mandatory for a correct description of the eigenfrequencies. 

Expressed in terms of the base used for the description of the matrix elements (0.9), the corresponding eigenvectors are... 

Discriminant analysis evaluates the distance between individual points and several centroids hypothesized to exist in the hyperspace defined by elemental concentrations. Davis (28) provides a clear and concise description of the algebra involved in two-group and multiple-group discriminant analysis, showing that discriminant functions are equivalent to the eigenvectors of W-1B, where W1 is the inverse of the within-group sums of products matrix, and B is the between-group sums of products matrix. The Mahalanobis distances from an unknown point to each of the alternative centroids... 

As previously discussed, a description of the temporal evolution of a system is accomplished by stating the relationship between eigenvectors associated with different times or, in other words, by exhibiting the transformation function in eqn (8.71). One may expect that the quantum dynamical laws will find their proper expression in terms of the transformation function and we now present Schwinger s development (1951) of a differential formulation of this type. 

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