Eigenvalue-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... 

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... 

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). 

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... 

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 ... 

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. 

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