Row matrix

If = 1 then we have a matrix consisting of a single column of terms and this referred to as a column matrix. If m — I then the matrix is called a row matrix. 

We may regard these two results as matrix relations the left side of each equation is a row matrix, the index running along the row the quantities or then form matrices, rows and columns... 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

The outer product of two vectors can be thought of as the matrix product between a single-column matrix with a single-row matrix ... 

If m equals n, the array is a square matrix of order n. If we have m = I, then the matrix has only one row and is known as a row matrix. On the other hand, if we have — 1, then the matrix consists of one column and is called a column matrix. 

Let us begin by representing a row matrix M = (1,2,3) in column space as shown in Figure 14-1. Note that the row vector M = (1,2, 3) projects onto the plane defined by columns 1 and 2 as a point (1, 2) or a vector (straight line) with a Cx direction angle (a) equal to... 

Since the wave function may be interpreted as a column vector with four components, may be defined to be a row matrix with components and which satisfies the adjoint equation... 

If a matrix contains only one row, it is called a row matrix or a row vector. The matrix B above is an example of a 1x3 row vector. Similarly, a matrix containing only one column is known as a column matrix or column vector. The matrix C above is a 6x1 column vector. One use of vectors is to represent the location of a point in... 

This vector is sometimes represented as a one-row matrix or a column vector. Usually, because of context, there in no confusion that stems from these alternative representations. More discussion on this point can be found in Appendix A.) As long as the dimensions are sufficiently small, the orthogonal (z, r, 9) coordinate system becomes sufficiently close to a cartesian system. In fact the arguments that follow are identical to those made in a cartesian setting. The planes that are formed by the intersection of A with the coordinate axes have areas Az = nzA, Ar = nrA, and Aq = ngA. These four planes form a tetrahedron. The discussion that follows considers the limit of vanishingly small dimensions, that is, shrinking the tetrahedron to a point. 

In Section 1.2, we pointed out that an ordered set of n numbers can be regarded as defining a vector in an abstract /i-dimensional space. In line with this idea, a row matrix is often called a row vector, and a column matrix is often called a column vector. The general m by n matrix (2.1) can be regarded as either a set of m ordered row vectors or a set of n ordered column vectors. 

Next we discuss multiplication of one matrix by another. Before considering the general case, we define the matrix product of a row matrix and a column matrix. If A is a row matrix and B is a column matrix having the same number of elements as A, then the product AB is a 1 by 1 matrix, calculated according to... 

A matrix containing only one row is called a row matrix it has the form... 

For a square matrix, taking the transpose amounts to reflecting the elements about the principal diagonal. A symmetric matrix is equal to its transpose A = A. The transpose of a row matrix is a column matrix, and vice versa. 

A type of matrix that is of considerable importance is the one-column matrix. To have the convenience of writing such a matrix all on one line, it is sometimes written out horizontally but enclosed in braces,, so as to distinguish it from a one-row matrix, which is normally written on one line in square brackets. The chief significance of the column matrix, at least for our purposes, is that it affords a way of representing a vector. Indeed it is sometimes actually called a vector. 

The question arises Can objects be defined in S which are represented by the 1 x n row matrix rf Well, not quite in S itself. It turns out that to any vector x in S there corresponds a so-called bra functional, written as (, with the defining property... 

Specifically, in Chapter 3 we create a surface for a transcendental function /(a, y) as an elevation matrix whose zero contour, expressed numerically as a two row matrix table of values, solves the nonlinear CSTR bifurcation problem. In Chapter 6 we investigate multi-tray processes via matrix realizations in Chapter 5 we benefit from the least squares matrix solution to find search directions for the collocation method that helps us solve BVPs and so on. Matrices and vectors are everywhere when we compute numerically. That is, after the laws of physics and chemistry and differential equations have helped us find valid models for the physico-chemical processes. 

Equation (7) describes the transformation of the set of basis vectors ei e2 e31 that are firmly embedded in configuration space and were originally coincident with fixed orthonormal axes x y z prior to the application of the symmetry operator R(n (3 7). In eq. (8) the column matrix x y z) contains the variables x y z, which are the components of the vector r = OP and the coordinates of the point P. In eq. (9) the row matrix (x y z contains the functions x y z (for example, the angle-dependent factors in the three atomic p functions px, py, pz). 

Example Al.2-1 If the row matrix (e contains the 3-D configuration space basis vectors ei e2 e3 and the column matrix r) contains the components xyz) of a vector r, then the matrix representative (MR) of the vector r is... 

In the Hiickel theory the three occupied MO, which determine the it bonds in benzene, PMO = 11, where 1 stands for the unit row matrix, read... 

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