Block-matrix notation

This system can be written in block-matrix notation... 

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 fact the relative coefficients within a set of symmetrically equivalent atoms, such as Cl, C3, C5, and Cg in para-benzosemiquinone, can be determined by group theory alone. The appropriate set of symmetrized orbitals, also listed in Table 1, can be obtained by use of character tables and procedures described in Refs. 3 to 6. In matrix notation, the symmetrized combinations are <1> = Up, where <1) and p are column vectors and U is the transformation matrix giving the relations shown in part (c) of Table 1. The transformation U and its transpose U can be used to simplify the solution of the secular matrix X since the matrix multiplication UXU gives the block diagonal form shown in part (6) of Table 1. 

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 ... 

Consider a block of N data samples s which are drawn from a short-term stationary AR process with parameters a. Equation 4.1 can be re-written in matrix/vector notation... 

There are two competing and equivalent nomenclature systems encountered in the chemical literature. The description of data in terms of ways is derived from the statistical literature. Here a way is constituted by each independent, nontrivial factor that is manipulated with the data collection system. To continue with the example of excitation-emission matrix fluorescence spectra, the three-way data is constructed by manipulating the excitation-way, emission-way, and the sample-way for multiple samples. Implicit in this definition is a fully blocked experimental design where the collected data forms a cube with no missing values. Equivalently, hyphenated data is often referred to in terms of orders as derived from the mathematical literature. In tensor notation, a scalar is a zeroth-order tensor, a vector is first order, a matrix is second order, a cube is third order, etc. Hence, the collection of excitation-emission data discussed previously would form a third-order tensor. However, it should be mentioned that the way-based and order-based nomenclature are not directly interchangeable. By convention, order notation is based on the structure of the data collected from each sample. Analysis of collected excitation-emission fluorescence, forming a second-order tensor of data per sample, is referred to as second-order analysis, as compared with the three-way analysis just described. In this chapter, the way-based notation will be arbitrarily adopted to be consistent with previous work. 

Besides the restrictions imposed on the orbital transformations to preserve spin symmetries, it is also useful to preserve spatial symmetry. This is done by allowing transformations only within sets of orbitals having the same symmetry properties and by not allowing these different sets of orbitals to mix. This restriction is accomplished by forcing the off-diagonal symmetry blocks of the K matrix, those labeled by spatial orbitals belonging to different symmetry types, to be zero. The notation required to label the symmetry species of the orbitals is somewhat cumbersome and will not be used except when explicitly required. 

The notation u = (u u ), v = (v v ) has been used. These simplifications are easy to introduce in any method based upon the utilization of cofactors. This is true whether the cofactors are computed directly, or indirectly as elements of the adjoint matrix of the overlap —provided only that the block diagonal form of is fully exploited. 

The Mie scattering efficiency database can be reorganised into a matrix (a block of numbers) in which each row is an individual Q curve. This matrix hereafter is denoted as Q, where the bold font indicates a matrix according to standard linear algebra notation. If our raw spectrum to be corrected comprises 2000 absorbance values, then the 1000 Q curve matrix will have 1000 rows and 2000 columns (a 1000x2000 matrix). 

For the two-dimensional irreducible representations (the matrix of symmetry operation diagonalized around the main diagonal in blocks of irreducible one-dimensional matrix) is used llie notation E" (do not be confused with the operation of symmetry identity ) and for the tri-dimen-sional irreducible representations the symbol 7 is used... 

First we note that the matrices T and X in (8.2.14) are rectangular, their columns referring only to the n occupied orbitals. With the notation of (8.3.1) the n X n matrix becomes TlXi, Xi being the first block of an extended X matrix in which Xj is identically zero. In terms of the full matrices, the quantity of interest is thus simply the leading diagonal block of... 

---