Matrices, Vectors, Scalars

It is helpful to distinguish matrices, vectors, scalars and indices by typographic conventions. Matrices are denoted in boldface capital characters (A), vectors in boldface lowercase (a) and scalars in lowercase italic characters (s). For indices, lower case characters are used (i). The symbol t indicates matrix and vector transposition (A4, a4). 

All chemical applications discussed later in this book will deal exclusively with real numbers. Thus, we introduce matrix algebra for real numbers only and do not include matrices formed by complex numbers. 

Sometimes it is helpful to specifically distinguish between row and column vectors. In such instances, we borrow Matlab s colon ( ) notation. A vector x 

Furthermore, every row of a matrix A can be seen as a row vector or sub matrix of A with the dimensions lxn, while eveiy column of A represents a column vector or sub matrix of the dimensions mxl. Thus, the second row of matrix A can be referred to as the row vector a2 , the third column of A as the column vector a 3, etc. With this notation it is generally possible to denote any sub matrix of A. For example, A2 4,3 6 is a matrix of dimensions 3x4 comprised of the elements of A that are within the rectangle defined by rows 2 to 4 and columns 3 to 6. Let s see how this is done in Matlab  

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Recall the colon ( ) notation as introduced in Chapter 2.1, Matrices, Vectors, Scalars. The first column of F, fi, contains m ones, while the second... 

Some functions operate on individual elements rather than rows or columns. For example, sqrt (W) results in a new matrix of dimensions identical with W containing die square root of all the elements. In most cases whether a function returns a matrix, vector or scalar is commonsense, but there are certain linguistic features, a few rather historical, so if in doubt test out the function first. 

Then the matrix from scalar products of the first p vectors is regular... 

Here, a few comments are in order. The matrix of derivative couplings F is antihermitian. The matrix of scalar couplings G is composed of an hermitian as well as an antihermitian part. Of course, the dressed kinetic energy operator —(1/2M)(V - - F) in our basic Eq. (10) is hermitian, as is also the case for the nonadiabatic couplings A in Eq. (9a). The latter follows immediately from the relation (lie). The notation (V F) is self evident from Eq. (lid). Since F is a vector matrix, it can be written as F = (Fi, F2,..., Fjv ), where the matrices Fq, are simply defined by their... 

Remember again that we have left out the unit dyads (xx, etc). In matrix notation the vector scalar product of eq. 1.2.4 becomes the multiplication of a row with a colutim matrix. 

The vector product and the scalar triple product can be conveniently written as matrix leterminants. Thus ... 

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

Matrix and tensor notation is useful when dealing with systems of equations. Matrix theory is a straightforward set of operations for linear algebra and is covered in Section A.I. Tensor notation, treated in Section A.2, is a classification scheme in which the complexity ranges upward from scalars (zero-order tensors) and vectors (first-order tensors) through second-order tensors and beyond. 

A few comments on the layout of the book. Definitions or common phrases are marked in italic, these can be found in the index. Underline is used for emphasizing important points. Operators, vectors and matrices are denoted in bold, scalars in normal text. Although I have tried to keep the notation as consistent as possible, different branches in computational chemistry often use different symbols for the same quantity. In order to comply with common usage, I have elected sometimes to switch notation between chapters. The second derivative of the energy, for example, is called the force constant k in force field theory, the corresponding matrix is denoted F when discussing vibrations, and called the Hessian H for optimization purposes. 

This provides an inductive, and a constructive, proof of the possibility of a triangular factorization of the specified form, provided only certain submatrices are nonsingular. For suppose first, that Au is a scalar, A12 a row vector, and A21 a column vector, and let Ln = 1. Then i u = A1U B12 — A12, and L2l and A22 axe uniquely defined, provided only Au = 0. But Au can be made 0, at least after certain row permutations have been made. Hence the problem of factoring the matrix A of order n, has been reduced to the factorization of the matrix A22 of order n — 1. 

Still another interpretation can be made by taking A22 to be a scalar, hence A21 a row vector and A12 a column vector. Suppose A1X has been inverted or factored as before. Then L21, R12, and A22 are obtainable, the two triangular matrices are easily inverted, and their product is the inverse of the complete matrix A. This is the basis for the method of enlargement. The method is to start with aai which is easily inverted apply the formulas to... 

Just as a known root of an algebraic equation can be divided out, and the equation reduced to one of lower order, so a known root and the vector belonging to it can be used to reduce the matrix to one of lower order whose roots are the yet unknown roots. In principle this can be continued until the matrix reduces to a scalar, which is the last remaining root. The process is known as deflation. Quite generally, in fact, let P be a matrix of, say, p linearly independent columns such that each column of AP is a linear combination of columns of P itself. In particular, this will be true if the columns of P are characteristic vectors. Then... 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

Lorentz invariant scalar product, 499 of two vectors, 489 Lorentz transformation homogeneous, 489,532 improper, 490 inhomogeneous, 491 transformation of matrix elements, 671... 

A matrix is defined as an ordered rectangular arrangement of scalars into horizontal rows and vertical columns (Section 9.3). On the one hand, one can consider a matrix X with n rows and p columns as an ordered array of p vectors of dimension n, each of the form ... 

Fig. 29.6. Schematic illustration of four types of special matrix products the matrix-by-vector product, the vector-by-matrix product, the outer product and the scalar product between vectors, respectively from top to bottom.

The bracket (bra-c-ket) in

) provides the names for the component vectors. This notation was introduced in Section 3.2 as a shorthand for the scalar product integral. The scalar product of a ket tp) with its corresponding bra (-01 gives a real, positive number and is the analog of multiplying a complex number by its complex conjugate. The scalar product of a bra tpj and the ket Aj>i) is expressed in Dirac notation as (0yjA 0,) or as J A i). These scalar products are also known as the matrix elements of A and are sometimes denoted by Ay. 

The scalar product of the vectors x and y when expressed in matrix notation is... 

The eigenvalue problem can be described in matrix language as follows. Given a matrix ff, determine the scalar quantities X and the nonzero vectors U which satisfy simultaneously file equation... 

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