Cross-products

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

Equations (169) and (171), together with Eqs. (170), fomi the basic equations that enable the calculation of the non-adiabatic coupling matrix. As is noticed, this set of equations creates a hierarchy of approximations starting with the assumption that the cross-products on the right-hand side of Eq. (171) have small values because at any point in configuration space at least one of the multipliers in the product is small [115]. 

The tensor (cross) product of a tensor with a vector is found as follows... 

The torque is given by the vector cross product of the vectors pi and Hq. 

Sometimes potential energy surfaces are plotted with skewed axes that is, the Tab 2nd tbc axes meet at an angle less than 90°. This is done so that the relative kinetic energy of the three-body system can be represented by the motion of a single point over the surface. In order to achieve this condition it is necessaiy that the cross-product terms in the kinetic energy drop out. The calculations have been described - Because our use of potential energy surfaces is qualitative,... 

Gradient operator Laplace operator Dot product Cross product Divergence operator Curl operator Vector transposition Complex conjugate... 

Following usual conventions, repeated indices indicate summation and fy denotes df/dXj. The permutation S5mibol is used to present the vector cross product in indicial notation. Due to the anisotropic nature, traction and body couples can exist, and thus the angular momentum equation must be considered. For purely viscous fluids this equation says simply that the deviatoric stresses are symmetric. 

For R = benzylamine (B) and R — cyclohexylamine (C) the amounts of BB and CC and of the BC cross product formed as the relative concentration of the reactants varies is shown in Figure 3. Clearly whilst for an approximately 50 50 reactant mixture the amount of CC formed has dropped to practically zero that of the cross product (BC) is still hi. The interpretation suggests that the less baisic atmine is more readily intercalated auid essentially remains urprotonated whilst the more basic amine, althou present in much smaller concentration, removes nearly all of the available protons. Hence the major reaction is between protonated B and non-protonated C, yielding BC (23,23). 

It can be shown that all symmetric matrices of the form X X and XX are positive semi-definite [2]. These cross-product matrices include the widely used dispersion matrices which can take the form of a variance-covariance or correlation matrix, among others (see Section 29.7). 

Note that there is a linear dependence in X which is trtmsmitted to the matrix of cross-products A ... 

A theorem, which we do not prove here, states that the nonzero eigenvalues of the product AB are identical to those of BA, where A is an nxp and where B is a pxn matrix [3]. This applies in particular to the eigenvalues of matrices of cross-products XX and X which are of special interest in data analysis as they are related to dispersion matrices such as variance-covariance and correlation matrices. If X is an nxp matrix of rank r, then the product X X has r positive eigenvalues in A and possesses r eigenvectors in V since we have shown above that ... 

From the 4x2 matrix X of our previous illustration we already derived V and from the eigenvalue decomposition of the 2x2 cross-product matrix X X ... 

In a similar way we can derive the eigenvalue decomposition of the corresponding 4x4 cross-product matrix XX ... 

After preprocessing of a raw data matrix, one proceeds to extract the structural features from the corresponding patterns of points in the two dual spaces as is explained in Chapters 31 and 32. These features are contained in the matrices of sums of squares and cross-products, or cross-product matrices for short, which result from multiplying a matrix X (or X ) with its transpose ... 

For the 4x3 matrix X in the previous illustration in this section, we derive the cross-product matrices ... 

A special form of cross-product matrix is the variance-covariance matrix (or covariance matrix for short) Cp, which is based on the column-centered matrix Yp derived from an original matrix X ... 

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

Orthogonal rotation produces a new orthogonal frame of reference axes which are defined by the column-vectors of U and V. The structural properties of the pattern of points, such as distances and angles, are conserved by an orthogonal rotation as can be shown by working out the matrices of cross-products ... 

The decompositions of C and into U, V and A are illustrated in the following example, which makes use of the previously computed results. For the cross-products between wind directions, we obtain ... 

For the cross-products between trace elements we find that ... 

In the previous subsection, we have described S and L as containing the coordinates of the rows and columns of a data table in factor-space. Below we show that, in some cases, it is possible to graphically reconstruct the data table and the two cross-product matrices derived from it. It is not possible, however, to reconstruct at the same time the data and all the cross-products, as will be seen. We distinguish between three types of reconstructions. 

In the case where a equals 1 we can reconstruct the diagonalized cross-product matrix A ... 

In the current example using a = 1, we can reconstruct the sums of cross-products C between the wind directions from their scores S ... 

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