Eigenvector problem

Alternatively, es can be found from either a normal-mode or a quasi-harmonic-mode decomposition [118] by solving an eigenvalue-eigenvector problem... 

Rotation of the translated factor axes is an eigenvalue-eigenvector problem, the complete discussion of which is beyond the scope of this presentation. It may be shown that there exists a set of rotated factor axes such that the off-diagonal terms of the resulting S matrix are equal to zero (the indicates rotation) that is, in the translated and rotated coordinate system, there are no interaction terms. The relationship between the rotated coordinate system and the translated coordinate system centered at the stationary point is given by... 

In the case where the data matrix M is square, the SVD reduces to the classic eigenvector problem. It is important to note that the vectors in are ordered in descending order of their contribution to the total signal variance of the system. 

Minimization of this energy with the constraint that remain normalized (<I> = 1 = Ejj QCj << >il< >j>) gives rise to a so-called secular or eigenvalue-eigenvector problem ... 

Solution of eigenvalue-eigenvector problems, where we find the eigenvalue x and the eigenvector u of the square symmetric matrix A such that... 

This is a quantity which can be easily constructed given a set of molecular orbitals (the coefficients C ) and a precalculated set of atomic orbital integrals. At this point, the Hartree-Fock equations have been reduced to a matrix eigenvector problem, FC = SCe, but not in a computationally convenient form. Following the analysis leading to equation 84, we first define the transformed Fock matrix as... 

In order to find extrema of E( ui ), subject to the normalization condition, standard moves known as the Lagrange multipliers method are applied, which readily lead us to the well-known form of the generalized matrix eigenvalue/eigenvector problem ... 

As mentioned earlier, the resolvent is a tool allowing one to formally write down the solution of an eigenvalue/eigenvector problem. It is also useful for developing perturbation expansions, which, as we saw previously, require somewhat tedious work when done in terms of vectors (wave functions). 

With these notations the Hartree-Fock problem acquires the form of an eigen-value/eigenvector problem ... 

Rank annihilation methods employ eigenvalue-eigenvector analyses for direct determination of analyte concentration with or without intrinsic profile determination. With the exception of rank annihilation factor analysis, these methods obtain a direct, noniterative solution by solving various reconstructions of the generalized eigenvalue-eigenvector problem. 

Alternating least squares (ALS) methods are both slower, due to their numeric intensity, and more flexible than eigenvalue-eigenvector problem-based methods for solving Equation 12.1a and Equation 12.1b. The basic PARAFAC model of Equation... 

The problem of finding the rotation matix that will "diagonalize" some symmetric, Hermitian, or unitary matrix A can be recast as an eigenvalue-eigenvector problem We seek the characteristic solutions to the problem... 

This is an eigenvector problem and hence the frequencies can be calculated by diagonalizing the dynamical matrix, although this procedure must be repeated for all of the wavevectors considered in the sampling method. 

A is the associated eigenvalue. The eigenvector problem can be reformulated as follows ... 

Evaluation of the energy in the time-independent Schrodinger equation requires the solution of an eigenvalue-eigenvector problem [22]. For an electronic wave function satisfying Eq. (2.9), an eigenvector— the total electronic energy— wiU be found. A possible poly electronic wave function for n electrons could have the form of a Hartree product ... 

The eigenvalue/eigenvector problem arises in the determination of the values of a constant X for which the following set of n linear algebraic equations has nontrivial solutions ... 

This variational problem is equivalent to the eigenvalues and eigenvectors problem for the matrix ... 

In this standard eigenvalue-eigenvector problem, the /th column of the matrix C contains the LCAO coefficients for MO (pi and E is a diagonal matrix of eigenvalues e, corresponding to the energy of each MO. 

The core of ECVA is that reformulating the eigenvector problem in Equation (21) as a regression problem, it is possible to use PLS (See Chapter 4, Section 3.4) to deal with the possibility of ill-conditioned matrices. In the remainder of this paragraph, the theory behind ECVA will be briefly described first for the two-classes case and then for the multiple classes case, and an example of application will be provided. 

The largest size matrix for whieh the code developed here can be used is probably around a 20 by 20 matrix. This is sufficient to explore eigenvalue and eigenvector problems, but the reader is referred to the extensive literature on eigenvalue problems involving very large matrices. 

Matrix.eigenvalueO and Matrix.eigenvector() - Code for eigenvalue and eigenvector problems. 

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