Matrix nonsymmetric

In general, Q is a nonsymmetrical matrix whose components off the main diagonal are the coupling coefficients among the various fluxes involved. 

The principle of detailed balance which is also valid for the quantities fVqq- enables the diagonalization of the nonsymmetric matrix Wqq< with nonnegative elements ... 

A.7 Show that for a 2 X 2 symmetrical matrix, the eigenvalues must be real (do not contain imaginary components). Develop a 2 X 2 nonsymmetrical matrix which has complex eigenvalues. 

The poles of the frequency-dependent polarizability correspond to electronic excitations, occurring with an infinitesimal perturbation, i.e., h = 0. Substituting this into Eq. (2-17) leads to a nonsymmetric matrix eigenvalue problem ... 

On inspection it can be seen that J is a nonsymmetric matrix (j/, jj,). Therefore, the components of the rotation axis are ... 

The coefficients are obtained from rate constants and mutation frequencies and hence represent real quantities. The are the eigenvalues of a nonsymmetric matrix and need not be real. 

The square nonsymmetric matrix D contains elements which only depend on e and on cavity geometrical parameters (mutual distances of representative points, tesserae areas). It may be written in terms of two supplementary matrices B and A, such that ... 

The steps of the QR algorithm for calculating the eigenvalues and eigenvectors of a nonsingular nonsymmetric matrix A with real eigenvalues are the following ... 

Coomans et al. have applied ALLOC to the differentiation of pure milk from different species and mixtures. A nonsymmetrical loss matrix was used in two-category classifications ... 

So far, the preceding considerations have been very useful for characterization of states and for ordering of singlet-state energies relative to (T) with fixed values of K and K. Nevertheless, we could not receive any qualitative ideas about the shapes of energy surfaces as the biradicaloids develop from the normal molecules. For this purpose the introduction of overlap, which we have neglected so far, is necessary. The Hamiltonian matrix for nonorthogonal orbitals A and B for nonsymmetric biradicaloid is... 

Thus the excitation pulse can create a superposition of i), 2) consisting of two states of different reflection symmetry. The resultant superposition possesses no symmetry properties with respect to reflection [78]. We now show that the broken symmetry created by this excitation of nondegenerate bound states translates into a nonsymmetry in the probability of populating the degenerate , n, D ), , n, L ) continuum states upon subsequent excitation. To do so we examine the properties of the bound-free transition matrix elements ( , n, q de,g Ek) that enter into the probability of dissociation. Note first that although the continuum states , n, q ) are nonsymmetric with respect to reflection, we can define symmetric and antisymmetric continuum eigenfunctions , n, s ) and , n, a ) via the relations... 

As described above, the C2-nonsymmetric boron- and boron-tin-capped clathrochelate compounds can be prepared by a solid-phase synthesis on the element oxide surface that serves as both a matrix... 

Klein et al.216 have also shown that the (rwc)/ and (srrw)/ can be computed via eingevalues of matrix M. However, matrix M is nonsymmetric unless the graph is regular. But in the general case the eigenvalues of M are related to those of a symmetric matrix, since M can be given as ... 

It is clear from this small selection of data that the matrix of multicomponent diffusion coefficients may be a complicated function of the composition of the mixture. The matrix [D] is generally nonsymmetric, except for two special cases identified below. The cross coefficients (/ = = k) can be of either sign indeed it is possible to alter the sign of these cross-coefficients by altering the numbering of the components. 

Both these normalized matrices pAS and ES ) are called stochastic because a —> stochastic matrix is a square matrix with nonnegative values and the property that the sum of the elements in each row (or column) is equal to 1. Moreover, these matrices, unlike the matrices they are derived from, are nonsymmetric. 

Here the electronic wavefunctions can be taken as nonsymmetrized or symmetrized. The matrix elements are identical in either case. The ideal starting point would be a basis set, i and 2, that minimizes the values of the Eq. (3.3.1) off-diagonal matrix elements. Unfortunately, it is not possible to find solutions of the electronic Hamiltonian for which both terms of Eq. (3.3.1) are zero. Two possible types of deperturbed or zeroth-order electronic functions may be defined (see also Table 3.1) ... 

This equation implies that A(2>) is in general nonsymmetrical. As an example for the digraph the fdlonring adjacency matrix may be derived ... 

We notice that C has nonzero matrix elements only in a nonsymmetric irreducible representation of 0(4) This also explains why the use... 

---