Matrices and Matrix Operations

This section will briefly review some of the basic matrix operations. It is not a comprehensive introduction to matrix and linear algebra. Here, we will consider the mechanics of working with matrices. We will not attempt to explain the theory or prove the assertions. For a more detailed treatment of the topics, please refer to the bibliography. 

For our purposes, we can simply consider a matrix as a set of scalars organized into columns and rows. For example, consider the matrix A  

The following statements about A (or, as it is sometimes written, (A] are true  

Each column is a column vector containing 5 elements. 

It contains positive and negative values. (Most matrices encountered in chemometrics will contain only positive values.) 

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Equation (B4.2.2) can be written in a more general way by using matrices and matrix operators ... 

One of the air of multivariate analysis is to reveal patterns in the data, whether they are in the form of a measurement table or in that of a contingency table. In this chapter we will refer to both of them by the more algebraic term matrix . In what follows we describe the basic properties of matrices and of operations that can be applied to them. In many cases we will not provide proofs of the theorems that underlie these properties, as these proofs can be found in textbooks on matrix algebra (e.g. Gantmacher [2]). The algebraic part of this section is also treated more extensively in textbooks on multivariate analysis (e.g. Dillon and Goldstein [1], Giri [3], Cliff [4], Harris [5], Chatfield and Collins [6], Srivastana and Carter [7], Anderson [8]). 

The diagonal elements of a Hermitian matrix must be real. A real symmetric matrix is a special case of a Hermitian matrix. (The relation between Hermitian matrices and Hermitian operators will be shown in Section 2.3.)... 

Throughout this paper semi-heavy type will be used for matrices and matrix elements, gothic type for irreducible tensor operators (6, 15) and GUI sans for the ligand field operator, for proj ection operators, and for rotation operators (11). 

Two different approaches are commonly referred to in the literature when it comes to the automatic generation of reaction mechanisms. The first approach involves combinatorial algorithms based mainly on the pioneering work of Yoneda (1979). These generate the whole set of possible reactions by only taking into account the congruence of the electronic configuration of reactants and products. Bond electron matrices are used to represent the chemical species and matrix operators describe all the possible reactions. 

Matrices and mathematical operators have some things in common. There is a well-defined matrix algebra in which matrices are operated on and this matrix algebra is similar to operator algebra. Two matrices are equal to each other if and only if both have the same number of rows and the same number of columns and if every element of one is equal to the corresponding element of the other. The sum of two matrices is defined by... 

The exponential parametrization of a unitary operator is independent in the sense that there are no restrictions on the allowed values of the numerical parameters in the operator - any choice of numerical parameters gives rise to a bona fide unitary operator. In many situations, however, we would like to carry out restricted spin-orbital and orbital rotations in order to preserve, for example, the spin symmetries of the electronic state. Such constrained transformations are also considered in this chapter, which contains an analysis of the symmetry properties of unitary orbital-rotation operators in second quantization. We begin, however, our exposition of spin-orbital and orbital rotations in second quantization with a discussion of unitary matrices and matrix exponentials. 

From the time function F t) and the calculation of [IT], the values of G may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by = 2 . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. 

MATLAB supports every imaginable way that one can manipulate vectors and matrices. We only need to know a few of them and we will pick up these necessary ones along the way. For now, we ll do a couple of simple operations. With the vector x and matrix a that we ve defined above, we can perform simple operations such as... 

Since it is necessary to represent the various quantities by vectors and matrices, the operations for the MND that correspond to operations using the univariate (simple) Normal distribution must be matrix operations. Discussion of matrix operations is beyond the scope of this column, but for now it suffices to note that the simple arithmetic operations of addition, subtraction, multiplication, and division all have their matrix counterparts. In addition, certain matrix operations exist which do not have counterparts in simple arithmetic. The beauty of the scheme is that many manipulations of data using matrix operations can be done using the same formalism as for simple arithmetic, since when they are expressed in matrix notation, they follow corresponding rules. However, there is one major exception to this the commutative rule, whereby for simple arithmetic ... 

We can use elementary row operations, also known as elementary matrix operations to obtain matrix [g p] from [A c]. By the way, if we can achieve [g p] from [A c] using these operations, the matrices are termed row equivalent denoted by X X2. To begin with an illustration of the use of elementary matrix operations let us use the following example. Our original A matrix above can be manipulated to yield zeros in rows II and III of column I by a series of row operations. The example below illustrates this ... 

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). 

Another loose end is the relationship between the quasi-algebraic expressions that matrix operations are normally written in and the computations that are used to implement those relationships. The computations themselves have been covered at some length in the previous two chapters [1, 2], To relate these to the quasi-algebraic operations that matrices are subject to, let us look at those operations a bit more closely. 

As stated earlier, Matlab s philosophy is to read everything as a matrix. Consequently, the basic operators for multiplication, right division, left division, power (, /,, A) automatically perform corresponding matrix operations (A will be introduced shortly in the context of square matrices, / and will be discussed later, in the context of linear regression and the calculation of a pseudo inverse, see The Pseudo-Inverse, p.117). 

This modification for Boolean matrix multiplication permits use of the Boolean union operation (logical OR operation or logical sum) instead of regular multiplication and union operations. The Boolean union operation can be executed much faster on a digital computer. Experience has shown that performing the Boolean union of rows instead of the standard Boolean multiplication of matrices can reduce the computation time by as much as a factor of four. 

In the case of the strength matrices, the matrix elements are real for T-even operators and image for T-odd operators and thus we easily get... 

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