Small matrices

A diagonal matrix has nonzero elements only on the principal diagonal and zeros elsewhere. The unit matrix is a diagonal matrix. Large matrices with small matrices symmetrically lined up along the principal diagonal are sometimes encountered in computational chemistry. 

The paper-and-pencir method of eigenvector decomposition can only be performed on small matrices, such as illustrated above. For matrices with larger dimensions one needs a computer for which efficient algorithms have been designed (Section 31.4). 

The scaling laws of FD are dominated by the recursion because diagonalization of small matrices is relatively inexpensive. However, because one must store multiple filtered vectors along the recursion, FD could be a burden for large systems. 

PLS is best described in matrix notation where the matrix X represents the calibration matrix (the training set, here physicochemical parameters) and Y represents the test matrix (the validation set, here the coordinates of the odor stimulus space). If there are n stimuli, p physicochemical parameters, and m dimensions of the stimulus space, the equations in Figure 6a apply. The C matrix is an m x p coefficient matrix to be determined and the residuals not explained by the model are contained in E. The X matrix is decomposed as shown in Figure 6b into two small matrices, an n x a matrix T and an a x p matrix B where a << n and a << p. F is the error matrix. The computation of T is such that it both models X and correlates with T and is accomplished with a weight matrix W and a set of latent variables U for Y with a corresponding loading matrix B. ... 

These are iterative procedures that allow the calculation and the Green s function matrix elements without explicit diagonalization of the Hamiltonian [18]. In the present case the Hamiltonian is factorized into double chains, and the renormalization method can be conveniently and efficiently applied, since its implementation simply requires the handling and the inversion of small matrices of rank two. For more details and elaboration of the renormalization procedures, see, for example, Ref. [23]. 

With this interpolation, the small matrices will be defined by... 

Although matrix multiplications, row reductions, and calculation of null spaces can be done by hand for small matrices, a computer with programs for linear algebra are needed for large matrices. Mathematica is very convenient for this purpose. More information about the operations of linear algebra can be obtained from textbooks (Strang, 1988), but this section provides a brief introduction to making calculations with Mathematica (Wolfram, 1999). 

A database on the thermodynamic properties of species of biochemical interest has been developed in Mathematica (7) as a package. In this package, BasicBiochemDataS (8), small matrices for 199 reactants (sums of species) contain the data at 298.15 K and zero ionic strength. There is a row in the matrix for each species that gives Af G ,Af W°, z,. ... 

When the notebook BasicBiochemDataS.nb was made, a package version BasicBiochemDataS.m was made automatically. The package consists of the Mathematica input without the text. When the 199 small matrices of species data or the 774 functions are needed, the command BiochemThermo BasicBiochem-DataS ... 

When the 199 small matrices of species data or the 11A functions are needed, the command BiochemThermo BasicBiochemData3 ... 

PCA and its application to chemometric problems have been reviewed by different authors [18,19], and a detailed description is beyond the scope of this article. However, we can introduce some of the basic concepts that will be used along this work. Basically, the PCA provides an approximations of the original multivariate description (the X matrix) in terms of two small matrices called scores (T) and loadings (P). Assuming that X is mean centered ... 

The newer PLS and PCR methods are two-factor-analysis based. Factor-analysis methods are used to factor the spectral data matrix into the product of two small matrices in order to simplify the data representation. This allows a more stable solution to be achieved when compared to the use of the ILS approach. 

The key idea of PCA is to approximate the original matrix X by a product of two small matrices - the score and loading matrices - according to... 

Crystal-field calculations based on the operator-equivalent method are performed by diagonalizing small matrices containing matrix elements of Hcep within a given state. The dimensions of these matrices are at most... 

This approximation resolves the computational difficulty encountered in the direct exact formulation that requires repeated computations of the solution of linear simultaneous algebraic equations and determinants of the matrices with huge dimensions. The efficiency in the approximated expansion is gained by the appreciation that the conditioning information can be truncated within one period of the system only. For linear systems, the expressions for the reduced-order likelihood function p(yi, yj, - - -, yNp W, C) and the conditional PDFs p(.yn 0, yn-Np, yn-Np+1, , y -i, C) are available since they are Gaussian and the correlation functions are known in closed forms regardless of the stationarity of the response. For stationary response, the method is very efficient in the sense that evaluation of all the conditional PDFs p(ynW, yn-Np,yn-Np+i,, y -i, C) requires the inverse and determinant of two relatively small matrices only. 

Another matrix, called the N matrix, is obtained by inverting the 1 - Q matrix. Matrix inversion can be done analytically for small matrices, but it is impractical to do for large realistic matrices that represent more complicated systems. A reasonable solution is available, however. Many spreadsheet programs in common use have the ability to numerically invert a matrix. This tool can be used to make quick work of previously time-consuming MTTF calculations. A numerical matrix inversion of the (1 - Q) matrix using a spreadsheet provides the following ... 

These conventional techniques proceeded by using an important property of symmetry functions, namely that functions of different symmetry do not interact. Using this fact, the Hamiltonian matrix for a polymer can be decomposed into a large number of small matrices, each of which transforms as a specific translational irreducible representation. The size of these matrices would correspond to the unit cell. Because they are constructed using complex symmetry functions, the matrices are complex ... 

Using these procedures only the small matrices B (t) (n n) are stored. The large hypermatrices B(t), are never stored in the BSM. 

---