Small component matrix elimination

Problems may occur if small principal components are eliminated in the process of reducing the number of significant principal components/singular values, because it might happen that one of the eliminated singular values is important for the prediction of a certain constituent concentration. The decomposition of the absorbance matrix A does not consider relationships between the concentrations and the absorbances. Therefore, the decomposition might be not optimal with respect to further use of the calibration model for prediction of concentrations in unknown samples. 

Since the X matrix is directly evaluated from the electronic solutions of the four-component Fock operator, which must therefore be diagonalized, the same pathologies regarding the negative-energy states discussed in chapters 8 and 10 pose a caveat. However, if a four-component calculation must be carried out before the two-dimensional operator can be evaluated (as in the X2C case), the projection to electronic states by elimination of the small component in an exact two-component approach has no valid formal advantage (as the four-component variational solution for the 4-spinors already required (implicit) projection to the electronic solutions). 

After what has been derived in chapter 13, it is clear how this equation works. All difficulties with the elimination of the small component in chapter 13 stem from the fact that the relation between large and small component contains the potential energy operator V and also the energy. In the L6vy-Leblond equation this is avoided because the lower part of the matrix equation does not depend on either of them. This is most easily seen in split notation. 

Elimination (or isolation) of the small component provides a useful basis for discussion of the properties of the Dirae equation. In particular, in this section we want to develop the relationship between the large- and small-component basis sets. We write the matrix 2-spinor Dirac equation in the form of two coupled matrix equations,... 

An alternative to the operator approach is to start from the matrix equations (Filatov 2002). Then the elimination the small-component, the construction of the transformation and the transformed Fock matrix are all straightforward. There is no difficulty with interpretation because the inverse of a matrix is well defined. The matrix to be inverted is positive definite so it presents no numerical problems. The drawback of a matrix method is that the basis set for the small component must be used, at least to construct the potentials that appear in the inverse. In that case, the same number of integrals is required as in the full Dirac-Hartree-Fock method, and there is no reduction in the integral work or the construction of the Fock matrix. 

The theory for the matrix elimination of the small component follows closely the lines of the operator elimination of the small component, but there are some subtleties that arise from the fact that we have a metric matrix that is not the unit matrix. We will first perform the direct elimination, and subsequently discuss the Foldy-Wouthuysen transformation. 

Various approaches can be taken for constructing the U matrix. With PCR, a principal components analysis is used because PCA is an efficient method for finding linear combinations of variables that describe variation in the row space of R (See Section 4.2.2). With analytical chemistry data, it is usually possible to describe the variation in R using significantly fewer PCs than the number of original variables. This small number of columns effectively eliminates the matrix inversion problem. 

The basis for the performance of the alloy in VRLA batteries is corrosion of the lead-cadmium-antimony alloy to produce antimony in the corrosion layer of the positive grid, which thus eliminates the antimony-free eifect of pure lead or lead-calcium alloys. During corrosion, small amounts of antimony and cadmium present in the lead matrix are introduced into the corrosion product and thereby dope it with antimony and cadmium oxides. The antimony and cadmium give excellent conductivity through the corrosion product. The major component of the alloy, the CdSb intermetallic alloy, is not significantly oxidized upon float service, but may become oxidized in cycling service. 

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