Cross-product matrice

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). 

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 ... 

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. 

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. 

The trace of A is equal to the traces of the weighted cross-product matrices, which in turn are equal to the global weighted sum of squares c or global interaction 5 (eq. (32.27)) ... 

The second critical fact that comes from equation 70-20 can be seen when you look at the Chemometric cross-product matrices used for calibrations (least-squares regression, for example, as we discussed in [1]). What is this cross-product matrix that is often so blithely written in matrix notation as ATA as we saw in our previous chapter Let us write one out (for a two-variable case like the one we are considering) and see ... 

Note that since SVD is based on eigenvector decompositions of cross-product matrices, this algorithm gives equivalent results as the Jacobi rotation when the sample covariance matrix C is used. This means that SVD will not allow a robust PCA solution however, for Jacobi rotation a robust estimation of the covariance matrix can be used. 

The procedure depicted in Figure 3 is the original and more general MEDA algorithm. Nevertheless, provided KDR is the missing data estimation technique, matrix can be computed from cross-product matrices following a more direct procedure. The value corresponding to the element in the Z-th row and -th column of matrix in MEDA is equal to ... 

The relationship of the MEDA algorithm and cross-product matrices was firstly pointed out by Arteaga (28) and it can also be derived from the original MEDA paprer (11). Equation (2) represents a direct and fast procedure to compute MEDA, similar in nature to the algorithms for model fitting from cross-product matrices, namely the eigendecomposition (ED) for PCA... 

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 ... 

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 ... 

An important aspect of latent vectors analysis is the number of latent vectors that are retained. So far, we have assumed that all latent vectors are involved in the reconstruction of the data table (eq. (31.1)) and the matrices of cross-products (eq. (31.3)). In practical situations, however, we only retain the most significant latent vectors, i.e. those that contribute a significant part to the global sum of squares c (eq.(31.8)). 

From our previous chapter defining the elementary matrix operations, we recall the operation for multiplying two matrices the i, j element of the result matrix (where i and j represent the row and the column of an element in the matrix respectively) is the sum of cross-products of the /th row of the first matrix and the y th column of the second matrix (this is the reason that the order of multiplying matrices depends upon the order of appearance of the matrices - if the indicated ith row and y th column do not have the same number of elements, the matrices cannot be multiplied). 

The M 1 matrices specific for higher point groups are obtained by omission of symmetry-forbidden columns in the full 15 x 15 matrix. This leads to rows with zero elements for the nonallowed cross products between d orbitals, which are subsequently omitted to recover a reduced matrix. The matrix for the point group D4h = 4/m mm is shown as an example in Table 10.2. 

Using the matrices of sums of squares and cross products immediately preceding Section 3.2.3, compute the coefficients in the multiple regression of real investment on a constant, real GNP and the interest rate. Compute R1. The relevant submatrices to be used in the calculations are... 

There is also a relation between polar unit vectors, boost generators, and electric fields. An electric field is a polar vector, and unlike the magnetic field, cannot be put into matrix form as in Eq. (724). The cross-product of two polar unit vectors is however an axial vector k, which, in the circular basis, is e<3>. In spacetime, the axial vector k becomes a 4 x 4 matrix related directly to the infinitesimal rotation generator /3) of the Poincare group. A rotation generator is therefore the result of a classical commutation of two matrices that play the role of polar vectors. These matrices are boost generators. In spacetime, it is therefore... 

The cross-product xj therefore becomes a commutator of matrices... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

More recently, it has been shown, in particular by Fowkes and co-workers [2,6,7], that electron acceptor and donor interactions, according to the generalized Lewis acid-base concept, could be a major type of interfacial forces between two materials. This approach is able to take into account hydrogen bonds which are often involved in adhesive joints. Inverse gas chromatography at infinite dilution for example is a well adapted technique [8-10] for determining the acid-base characteristics of fibres and matrices. Retention data of probes of known properties, in particular their electron acceptor (AN) and donor (DN) numbers according to Gutmann s semi-empirical scale [11], allow the determination of acid-base parameters, and Kj), of fibre and matrix surfaces. It becomes then possible to define a "specific interactions parameter" A at the fibre-matrix interface, as the cross-product of the coefficients and Kq of both materials [10,11] ... 

Here h and O3 denote the 3 x 3-dimensional identity and zero matrices, while the / operator refers to the cross-product tensor associated with Fk — Fk I, where Fk is the position vector that defines the reference frame for rigid body k. The spatial velocity, V i, can be computed from the following relation ... 

If, however, we use a set of coordinates (which are not necessarily normal coordinates) that transform under the symmetry operations of the group to which the molecule belongs in the manner indicated by the matrices of the irreducible representations, then all cross products of the type QiQj, where Qi and Qj belong to different irreducible representations, will vanish this choice of coordinates will thus greatly simplify the solution of equation 2-55. We shall return to this point briefly a little later first we wish to derive the selection rules for optical transitions between the various possible vibrational states of polyatomic molecules. 

Field fortification (commonly referred to as field spiking) is the procedure used to prepare study sample matrices to which have been added a known amount of the active ingredient of the test product. The purpose for having field fortification samples available in a worker exposure study is to provide some idea of what happens to the test chemical under the exact environmental field conditions which the worker experiences and to determine the field storage stability of the test substance on or in the field matrix materials. Field fortifications do not serve the purpose of making precise decisions about the chemical, which can better be tested in a controlled laboratory environment. The researcher should not assume that a field fortification sample by its nature provides 100% recovery of the active ingredient at all times. For example, a field fortification sample by its very nature may be prone to cross-contamination of the sample from environmental contaminants expected or not expected to be present at the field site. 

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