Three-Dimensional Vector Algebra

A three-dimensional vector can be represented by specifying its components a,-, i = 1,2, 3 with respect to a set of three mutually perpendicular unit vectors as 

The vectors are said to form a basis and are called basis vectors. The basis is complete in the sense that any three-dimensional vector can be written as a linear combination of the basis vectors. However, a basis is not unique we could have chosen three different mutually perpendicular unit vectors, e, i = 1, 2, 3 and represented d as 

Given a basis, a vector is completely specified by its three components with respect to that basis. Thus we can represent the vector 5 by a column matrix as 

Given a vector a, we can find its component along ej by taking the scalar product of Eq. (1.1) with ej and using the orthonormality relation (1.7) 

We now define an operator 0 as an entity which when acting on a vector a converts it into a vector b 

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Now that we have seen how 3x3 matrices naturally arise in three-dimensional vector algebra and how they are multiplied, we shall generalize these results. A set of numbers [Aij that are in general complex and have ordered subscripts / = 1, 2,..., AT and 7 = 1, 2,..., M can be considered elements of a rectangular N x M) matrix A with N rows and M columns... 

We need to generalize the ideas of three-dimensional vector algebra to an iV-dimensional space in which the vectors can be complex. We will use the powerful notation introduced by Dirac, which expresses our results in an exceedingly concise and simple manner. In analogy to the basis ej in three dimensions, we consider N basis vectors denoted by the symbol i>, i = 1, 2,..., N, which are called ket vectors or simply kets. We assume this basis is complete so that any ket vector a> can be written as... 

In the course of this work there will be occasion to use vectors defined in more than three dimensions. Although an n-dimensional vector is difficult to visualize geometrically the algebraic definition is no more complicated than for three-dimensional vectors. 

The algebraic description of three-dimensional problems is in terms of U(4). The coset space for this case is a complex three-dimensional vector, We denote its complex conjugate by q. One can then introduce the canonical position and momentum variables q and p by the transformation... 

As a simple example of a Lie algebra consider the three-dimensional vector space with the unit basis vectors e2, e2, e3, each pointing along one of the mutually perpendicular coordinate axes. Define [et, e-] = et x e i, the usual... 

As has already been noted (Ch. 1), this system is closely connected with the three-dimensional Lie algebra so(3) of the rotation group SO(3). The point is that the space R (x, y, z) can be naturally identified with the space so(3) of skew-symmetric real matrices X = (x y). To this end, it suffices to put X12 = is = y) 23 = that is, to associate with the vector with coordinates (x, y, z) the matrix... 

Rules of matrix algebra can be appHed to the manipulation and interpretation of data in this type of matrix format. One of the most basic operations that can be performed is to plot the samples in variable-by-variable plots. When the number of variables is as small as two then it is a simple and familiar matter to constmct and analyze the plot. But if the number of variables exceeds two or three, it is obviously impractical to try to interpret the data using simple bivariate plots. Pattern recognition provides computer tools far superior to bivariate plots for understanding the data stmcture in the //-dimensional vector space. 

The matrices such as X and B in Eq. (33), which are composed of a single column, are usually referred to as vectors. In fact, the vectors introduced in Chapter 4 can be written as column matrices in which the elements are the corresponding components. Of course the vector X — [Xj ] in Eq. (33) is of dimension n, while those in Chapter 4 were in three-dimensional space. It is apparent that the matrix notation introduced here is a more general method of representing vector algebra in multidimensional spaces. This idea is developed further in Section 7.7. 

One important application of matrix algebra is formulating the transformations of points or vectors which define a geometrical entity in space. In ordinary three-dimensional space that involves three axes, any point is located by means of three coordinates measured along these axes. Similarly... 

A useful feature of the algebraic representations of geometric quantities is the ease with which one can work in dimensions higher than three. Although it is difficult to visualize the angle between two five-dimensional vectors, there is no particular problem involved in taking the dot product between two vectors of the form (xi,X2,xs,X4,X5). 

Algebraic description of symmetry operations is based on the following simple notion. Consider a point in a three-dimensional coordinate system with any (not necessarily orthogonal) basis, which has coordinates x, y, z. This point can be conveniently represented by the coordinates of the end of the vector, which begins in the origin of the coordinates 0, 0, 0 and ends at x,y, z. Thus, one only needs to specify the coordinates of the end of this vector in order to fully characterize the location of the point. Any symmetrical transformation of the point, therefore, can be described by the change in either or both the orientation and the length of this vector. 

For the following derivations it may be convenient to recollect a few equations from the algebra of vectors in three-dimensional space ... 

The quantities in the two-dimensional list are called matrix elements. Each matrix element has two subscripts, one for the row and one for the column. The brackets written on the left and right are part of the notation. If a matrix has the same number of rows as columns m = n), it is a square matrix. A vector in ordinary space can be represented as a list of three Cartesian components, which is a matrix with one row and three columns. We call this a row vector. A vector can also be represented by a column vector with three rows and one column. We can also define row vectors and column vectors with more than three elements when they apply to something other than ordinary space. Just as there are types of algebra for scalars, vectors, and operators, there is a well-defined matrix algebra. 

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