Symmetrical matrix

Write out the coefficients of the butadienyl system, as they are produced by program MOBAS, in matrix form. Is the matrix symmetric If not, can it be made symmetric by exchanging rows only ... 

Note that transformation by a general coordinate transformation matrix leaves the quadrupole matrix symmetrical, i.e., Py = Pp and with zero trace,... 

Zero matrix Diagonal matrix Identity matrix Symmetric matrix... 

Exercise. For one-step processes W is a tridiagonal matrix. With the aid of (3.8) a similarity transformation can be constructed which makes the matrix symmetric, as in (V.6.15). Prove in this way that any finite one-step process has a complete set of eigenfunctions, and that its autocorrelation function consists of a sum of exponentials.510... 

The anisotropic displacement parameters can also be represented in a format of a tensor, Ty, i.e. a square matrix symmetrical with respect to its principal diagonal. For By it is given as... 

To emphasize how the Metropolis scheme works in practice, let us return once more to the hard-disk example. Here the matrix am represents the probability of moving a particle to a new trial positions, which is achieved by the generation of two random numbers for displacements in the x and directions. To make the underlying matrix symmetric, the displacements are calculated from dmlJ2 — 1), where [Pg.6]

In order to keep the Hamiltonian matrix symmetric we have to introduce a normalization condition for the radial function Eq. (12)... 

The results using SRIM procedure have shown that a channel distribution system in combination with equidistant multiple inlet holes facilitates the impregnation of reinforcement fiber structure with an elastomeric polymer matrix. Symmetric arrangement of the channels gives symmetric polymer flow into the cavity. This makes fastening of the reinforcement structure to the mold during resin injection unnecessary in order to prevent dislocation of the fiber mat inside the cavity. 

A remark on flux conservation. In the traditional inelastic scattering theory, one shows that the logarithmic derivative matrix Z is a symmetric matrix, because of the symmetry of coupling matrices. Moreover, the Wronskian of asymptotic functions is unity and it is easy to show that the K-matrix is symmetric and the S-matrix symmetric and unitary. In reactive collisions, the symmetry of the logarithmic derivative matrix is destroyed by the transformation (3.24) because U. Also, the matrix Wronskian of the asymptotic... 

Subroutine SYMINV. This subroutine inverts a symmetric matrix. 

This definition implies that the direction of the vector d was taken into account and that the coefficient c(i,j,fd)sndc(j,i,f,-d) may be different. So, the symmetrical matrix consists of a two step operation ... 

Another useful property of the S matrix is that it is symmetric. This property follows from conservation of the fluxlike expression... 

As in the case of the propagator, we shall be applying a symmetrical version of the Trotter fomuila [48] to the high-temperature density matrix... 

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The ti eatment of the Jahn-Teller effect for more complicated cases is similar. The general conclusion is that the appearance of a linear term in the off-diagonal matrix elements H+- and H-+ leads always to an instability at the most symmetric configuration due to the fact that integrals of the type do not vanish there when the product < / > / has the same species as a nontotally symmetiic vibration (see Appendix E). If T is the species of the degenerate electronic wave functions, the species of will be that of T, ... 

The implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

The question remains how to evaluate exp(—iTH(qo)/(2 )) i/ i while retaining the symmetric structure. In Sec. 4.2 we will introduce some iterative techniques for evaluating the matrix exponential but the approximative character of these techniques will in principle destroy the symmetry. 

When considering the construction of exactly symmetric schemes, we are obstructed by the requirement to find exactly symmetric approximations to exp(—ir/f/(2fi,)). But it is known [10], that the usual stepsize control mechanism destroys the reversibility of the discrete solution. Since we are applying this mechanism, we now may use approximations to exp —iTH/ 2h)) which are not precisely symmetric, i.e., we are free to take advantage of the superior efficiency of iterative methods for evaluating the matrix exponential. In the following, we will compare three different approaches. 

Here, M is a constant, symmetric positive definite mass matrix. We assume without loss of generality that M is simply the identity matrix I. Otherwise, this is achieved by the familiar transformation... 

We assume that A is a symmetric and positive semi-definite matrix. The case of interest is when the largest eigenvalue of A is significantly larger than the norm of the derivative of the nonlinear force f. A may be a constant matrix, or else A = A(y) is assumed to be slowly changing along solution trajectories, in which case A will be evaluated at the current averaged position in the numerical schemes below. In the standard Verlet scheme, which yields approximations y to y nAt) via... 

As can be seen in Figure 2-13, the diagonal elements of the matrix are always zero and it is symmetric around the diagonal elements (undirected, unlabeled graph). Thus, it is a redundant matrix and can be reduced to half of its entries (Figure 2-14b. For clarity, all zero entries are omitted in Figures 2-14b-d. 

Given a graph G with n vertices, the adjacency matrix A is a square nxn symmetric matrix (Eq. 9 ). 

As both the diagonal matrix DEG and the adjacency matrix A are symmetric it follows that the Laplacian matrix (Eq. (12)) is also a symmetric one. 

The distance matrix D of a graph G with n vertices is a square n x n symmetric matrix as represented by Eq. (13), where is the distance between the vertices Vi and Vj in the graph (i.e., the number of edges on the shortest path). 

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