Eigenvector problem, solutions

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

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. 

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 ... 

However, viewed as an eigenvalue problem, principal components have some features of interest. First of all, we note that eigenvectors are solutions to the equation ... 

As the outcome, a list of eigenvalues A and the modal matrix H with the pairwise conjugate complex eigenvectors of the general problem is obtained. Due to the normahzation of the eigenvectors, the solution has to be matched to the initial conditions. This is done when the overall solution is assembled and for this purpose the vector h is provided. The homogeneous solution thus takes the following form ... 

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

Molecules, in general, have some nontrivial symmetry which simplifies mathematical analysis of the vibrational spectrum. Even when this is not the case, the number of atoms is often sufficiently small that brute force numerical solution using a digital computer provides the information wanted. Of course, crystals have translational symmetry between unit cells, and other elements of symmetry within a unit cell. For such a periodic structure the Hamiltonian matrix has a recurrent pattern, so the problem of calculating its eigenvectors and eigenvalues can be reduced to one associated with a much smaller matrix (i.e. much smaller than 3N X 3N where N is the number of atoms in the crystal). 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the F >v matrix elements depend on the Cv,i LCAO-MO coefficients which are, in turn, solutions of the so-called Roothaan matrix Hartree-Fock equations- Zv F >v Cv,i = Zv S v Cvj. One should also note that, just as F (f>j = j (f>j possesses a complete set of eigenfunctions, the matrix Fp,v, whose dimension M is equal to the number of atomic basis orbitals used in the LCAO-MO expansion, has M eigenvalues j and M eigenvectors whose elements are the Cv>i- Thus, there are occupied and virtual molecular orbitals (mos) each of which is described in the LCAO-MO form with CV)i coefficients obtained via solution of... 

Because of the spin-spin coupling term, (8.41) is not separable into the sum of Hamiltonians for the individual nuclei, and the corresponding Schrodinger equation is not separable. To deal with the problem, we shall use the method of expanding the unknown wave functions in terms of a known complete set of functions. The eigenvalues and eigenfunctions (eigenvectors) are obtained as the solutions of (2.68) [or (2.77)] and (2.67). 

Each DCT(q) block appears ca times along the diagonal. The eigenvalues of DCT(q) are and their degeneracy is 1(a), the dimension of the IR a. This completes the solution to the problem of finding the frequencies and the eigenvectors of the dynamical matrix D(q), except for a consideration of extra degeneracies that may arise from time-reversal symmetry. 

The evolution of this determinant first yields the eigenvalues. The solution of the whole eigenvalue problem provides pairs of eigenvalues and eigenvectors. The mathematical algorithm is described in detail in [MALINOWSKI, 1991]. A simple example, discussed in Section 5.4.2, will demonstrate the calculation. The following properties of these abstract mathematical measures are essential ... 

The problem of finding a vector is usually solved by representing the required vector as an expansion with respect to some natural set of basis vectors. Following this method one can expand the vector of the n-th order correction to the k-th unperturbed vector- 44 n terms of the solutions b p (eigenvectors) of the unperturbed problem eq. (1.51) ... 

The residues theorem allows treating the resolvent as a formal solution of the eigenvector/eigenvalue problem. Indeed, taking a contour integral over any path Ct enclosing each of the poles one gets ... 

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