Matrix square block

This idea can easily be extended to more than two matrices to yield a matrix with non-vanishing elements in square blocks along the main diagonal and zeros elsewhere. Such a block-diagonal matrix (e.g. D = A B C) has the self-evident important properties ... 

A special case of matrix multiplication occurs when we deal with matrices having all nonzero elements in square blocks along the diagonal, such as the following two ... 

Images are represented by a matrix I of dimension 320 X 240. Previous to PCA, I is transformed by rearranging image blocks into columns. First, I is divided into p square blocks (if necessary, padded wifh zeros) of s s pixels second, a new matrix X with rows and p columns is formed by filling each column wifh the values in one of fhe blocks. Finally, PCA is applied to derive a lower-dimensional representation. Given a matrix X, and the desired new dimension, q, the PCA algorithm yields the matrix M (q rows and columns) and the vector B q components) needed to obtain a matrix Y q rows and p columns), according to ... 

These conclusions apply to the whole set of symmetry operations which form the point group of the molecule, so that it is possible to take advantage of the symmetry of the molecule to make the transformed matrix hu as sparse as possible. Further, by suitable numbering of the members of the basis (t>u, the matrix hu can be brought into a form in which it has (square) blocks of... 

Dropping the non-coupling hh and pp rows and columns we are left with a 3x3 matrix of square blocks of the same dimension which can be written in the following way ... 

Distribution of an nxn matrix across p processes, Pq — Pp-i, by (a) rows, (b) columns, and (c) square blocks. The one-dimensional distributions (a) and (b) can utilize at most n processes, whereas the two-dimensional block distribution (c) can distribute the matrix across up to rP processes. 

Fig. C.3. Reducible representation, block form, and irreducible representation. In the first row, the matrices F(Ri) are displayed that form a reilucible representation (eadi matrix corresponds to the symmetry operation Rj) the matrix elements are in general nonzero. The central row shows arepresentation F equivalent to the first one i.e., related by a similarity transformation (with matrix P). The new representation exhibits a block form i.e., in this particular case each matrix has two blocks of zeros that are identical in all matrices. The last row shows an equivalent representation F that corresponds to the smallest square blocks (of nonzeros) i.e., the maximum number of the blocks, of the form identical in all the matrices. Not only F, F, and F" are representations of the group, but also any sequence of individual blocks (as that shadowed) is a representation. Thus, F is decomposed into the four irreducible representations.

The dynamical matrix consists then of two square blocks composed of real elements (the translation-translation block and the libration-libration block) and two off-diagonal, generally rectangular blocks of pure imaginary elements (translation-libration interactions). Such a matrix can be transformed (Cochran and Pawley, 1964) into a real symmetric matrix of the same dimension by a transformation of the coordinates, which leaves all translational coordinates unchanged and transforms all librational coordinates u lk) as follows ... 

When internal coordinates are used, the F and G matrices and the secular determinant are square as in Fig. 14.5(a). When symmetry coordinates which transform as described above are used, the matrix obtained from UFU and the matrix obtained from UGU and the secular determinant I — IA = 0 all consist of smaller square blocks, one for each symmetry species, distributed along the diagonal as in Fig. 14.5(b). Since all terms in the nonshaded part of Fig. 14.5(b) are zero the problem factors into the symmetry species. 

Formulae such as Equation (4.3) are written using matrix notation by making the square block of the coefficients from the right-hand side into a matrix and writing the co-ordinates as column vectors ... 

U are one electron couplings between the two monomers and interactions proportional to overlap squares are neglected.This matrix becomes block diagonal by forming symmetric and anti symmetric combinations of the local transitions and of the CT states. An orthogonal transformation gives the coupling matrix for the even states... 

In order to determine the matrix thresholds, we present an expression of the coefficients dispersion that is related to the flattening of the cloud of the points around the central axis of inertia. The aim is to measure the distance to the G barycentre in block 3. So, we define this measure Square of Mean Distance to the center of Gravity as follow ... 

Partial least squares regression (PLS). Partial least squares regression applies to the simultaneous analysis of two sets of variables on the same objects. It allows for the modeling of inter- and intra-block relationships from an X-block and Y-block of variables in terms of a lower-dimensional table of latent variables [4]. The main purpose of regression is to build a predictive model enabling the prediction of wanted characteristics (y) from measured spectra (X). In matrix notation we have the linear model with regression coefficients b ... 

In addition, G and F matrix elements have been tabulated (see Appendix VII in Nakamoto 1997) for many simple molecular structure types (including bent triatomic, pyramidal and planar tetratomic, tetrahedral and square-planar 5-atom, and octahedral 7-atom molecules) in block-diagonalized form. MUBFF G and F matrices for tetrahedral XY4 and octahedral XY molecules are reproduced in Table 1. Tabulated matrices greatly facilitate calculations, and can easily be applied to vibrational modeling of isotopically substituted molecules. Matrix elements change, however, if the symmetry of the substituted molecule is lowered by isotopic substitution, and the tabulated matrices will not work in these circumstances. For instance, C Cl4, and all share full XY4 tetrahedral symmetry (point group Tj), but... 

The method which satisfies these conditions is partial least squares (PLS) regression analysis, a relatively recent statistical technique (18, 19). The basis of tiie PLS method is that given k objects, characterised by i descriptor variables, which form the X-matrix, and j response variables which form the Y-matrix, it is possible to relate the two blocks (or data matrices) by means of the respective latent variables u and 1 in such a way that the two data sets are linearly dependent ... 

The Jacobian of the system is a square matrix, but importantly, because the residuals at any mesh point depend only on variables at the next-nearest-neighbor mesh point, the Jacobian is banded in a block-tridiagonal form. Figure 16.10 illustrates the structure of the Jacobian in the form used by the linear-equation solution at a step of the Newton iteration,... 

The blocks of the block-tridiagonal structure correspond to the mesh, with each block being a square matrix with the dimension of the number of dependent variables at each mesh point (here the number of species, plus temperature, plus the mass flux). 

Partial least square (PLS) regression model describes the dependences between two variables blocks, e.g. sensor responses and time variables. Let the X matrix represent the sensor responses and the Y matrix represent time, the X and Y matrices could be approximated to few orthogonal score vectors, respectively. These components are then rotated in order to get as good a prediction of y variables as possible [25], Linear discriminant analysis (LDA) is among the most used classification techniques. The method maximises the variance between... 

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