Generalized eigenvalue problem

If the functions Oj are orthonormal, then the overlap matrix S reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form ... 

The variational Ritz procedure reduces the problem of solving (1) to solving the generalized eigenvalue problem ... 

In deriving this we have used the properties of the integrals Hij = //, and a similar result for Stj. Equation (1.14) is discussed in all elementary textbooks wherein it is shown that a Cy 0 solution exists only if the W has a specific set of values. It is sometimes called the generalized eigenvalue problem to distinguish from the case when S is the identity matrix. We wish to pursue further information about the fVs here. 

The generalized eigenvalue problem is unfortunately considerably more complicated than its regular counterpart when S = I. There are possibilities for accidental cases when basis functions apparently should mix, but they do not. We can give a simple example of this for a 2 x 2 system. Assume we have the pair of matrices... 

Our generalized eigenvalue problem thus depends upon three parameters, a, b, and s. Denoting the eigenvalue by W and solving the quadratic equation, we obtain... 

Here the N columns of U definejhe significant factors in the joint row space of R and R2, and the N columns of V define the joint column space. The generalized eigenvalue problem... 

Faber, K., On solving generalized eigenvalue problems using MATLAB,./. Chemom., 11, 87-91, 1997. 

The expansion column matrix A from Eq. (26) is the eigenvector of U(s), and the elements Snm of the overlap matrix S are given in Eq. (19). The obtained expression (26) is not an ordinary but a generalized eigenvalue problem involving the overlap matrix S due to the mentioned lack of orthogonality... 

Once the whole set uk,A /k is obtained by solving the generalized eigenvalue problem (26), the eigenfrequencies are deduced from the relation... 

Whatever the discretization in y is, the resulting equations are a generalized eigenvalue problem of the form AV = sBV A and B being two complex matrices, s the eigenvalue characterizing stability, and V the discretized vector of velocity, pressure, and extra-stress for each fluid. 

For the numerical study of the whole spectrum (for g R fixed), [79] uses a spectral tau-Chebychev discretization in y and the Arnold method (see [88]) to solve the generalized eigenvalue problem (see [89]). This numerical method is based on the orthonormalization of the Krylov space of the iterates of the inverse of the matrix A B. This method has been used more recently in [90]. It has been proven efficient in the stiff problems arising in the study of spectral stability of viscoelastic fluids. 

Note that we have elected to use the nomenclature A and B instead of the previously used D [Eq. (4)] to denote the different data matrices, and to clearly distinguish these matrices as proportional data sets. In addition, many of the previous DECRA derivations decompose the data matrices as A = CS where the transpose is utilized because in those descriptions the pure component spectral matrix is in a column representation (wfreq "comp)) d is the transpose of the definitions we presented in Section 2.1. Rearrangement of Eq. (46) leads to the generalized eigenvalue problem... 

Choosing the operators hi to be the state-transfer operators 4> ) (4 ol 4 o)(4 n would lead us back to the spectral representation, Eq. (23). In practical applications, however, the exact ground state of the system o) is replaced by some approximate wave function ), which is a linear combination of antisymmetrized products of molecular orbitals, so-called Slater determinants, while the operators hi replace one or more of the occupied molecular orbitals by virtual orbitals (excitations) in the Slater determinants or virtual orbitals by occupied orbitals (de-excitations). Approximations to the vertical electronic excitation energies E - Eq are then obtained by solving the generalized eigenvalue problem... 

Hy and Sy are tabulated for various distances between atom pairs up to 10 A, where they vanish. For any molecular geometry, these matrix elements are based on the distance between the atoms and then oriented in space by using the Slater-Koster sin/cos combination rules. Then, the generalized eigenvalue problem Equation 5.38 is solved and the first part of the energy can be calculated. It should be emphasized that this is a non-orthogonal TB scheme, which is more transferable due to the appearance of the overlap matrix. 

The optimization of the Hartree-Fock spin orbitals in Eq. (21) is a nonlinear minimization problem. By recasting Eq. (22) as a generalized eigenvalue problem, the optimization may be accomplished by repeated solution of the pseudo-eigenvalue... 

To solve for Ck, the generalized eigenvalue problem is used with the singular-value decomposition technique. The results of the problem indicate both the pure component response patterns x and y and the ratio of concentrations of the pure components to the standard response concentration. 

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