Matrices and Vectors

Campbell, H. G. An Introduction to Matrices, Vectors, and Linear Programming. Prentice-Hall, Englewood Cliffs, NJ (1977). 

These files store the workspace or variables created during a session. All matrices, vectors and scalars with unique names are saved. Many chemometricians exchange data in this format. The command save places all this information into a file called matlab.mat in the current working directory. Alternatively, you can use the Save Workspace item on die File menu. Normally you wish to save the information as a named file, in which case you enter the filename after the save command. The following code saves the results of a session as a file called mydata in the directory c results, the first line being dependent on the current working directory and requires you to have created this first ... 

This section presents the mathematical description of the NMR data sets and reviews some of the preprocessing and data manipulations that are employed to improve the performance of the subsequent chemometric analysis. Descriptions of the different chemometric methods involve the use of matrices, vectors and scalars, such that a brief note on the notation used in the subsequent sections is warranted. Throughout this... 

The distinction between matrices, vectors and arrays is important in the computer implementation of our equations because matrices are normally treated, accessed and manipulated as a whole since they are an internally connected collection of numbers while arrays are simply convenient places to keep sets of related but heterogeneous data orbital specifications, descriptions of atoms etc. 

An introduction to matrix algebra matrices, vectors and determinants... 

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

Throughout, unless otherwise stated, R and r will be used to represent the nuclear and electronic coordinates, respectively. Boldface is used for vectors and matrices, thus R is the vector of nuclear coordinates with components R. The vector operator V, with components... 

Symbols separated by commas represent equivalent recommendations. Symbols for physical and chemical quantities should be printed in italic type. Subscripts and superscripts which are themselves symbols for physical quantities should be italicized all others should be in Roman type. Vectors and matrices should be printed in boldface italic type, e.g., B, b. Symbols for units should be printed in Roman type and should remain unaltered in the plural, and should not be followed by a full stop except at the end of a sentence. References International Union of Pure and Applied Chemistry, Quantities, Units and Symbols in Physical Chemistry, Blackwell, Oxford, 1988 Manual of Symbols and Terminology for Physicochemical Quantities and Units, Pure Applied Chem. 31 577-638 (1972), 37 499-516 (1974), 46 71-90 (1976), 51 1-41, 1213-1218 (1979) 53 753-771 (1981), 54 1239-1250 (1982), 55 931-941 (1983) lUPAP-SUN, Symbols, Units and Nomenclature in Physics, PV ica 93A 1-60 (1978). 

X andjy are data matrices in row format, ie, the samples correspond to rows and the variables to columns. Some mathematical Hterature uses column vectors and matrices and thus would represent this equation as T = X. The purpose of rotation in general is to find an orientation of the points that results in enhanced understanding of the underlying chemical behavior of the system. 

Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vec tors in iT and for matrices that are analogous to the notion of length of a geometric vector. Let R denote the set of all vec tors with n components, x = x, . . . , x ). In dealing with matrices it is convenient to treat vectors in R as columns, and so x = (x, , xj however, we shall here write them simply as row vectors. 

The matrices F and M can be found from straightforward integration of (5.9) with the initial conditions being N linearly independent vectors. Then the quasienergy partition function equals... 

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. 

A matrix is a rectangular array of numbers, its size being determined by the number of rows and columns in the array. In this context, the primary concern is with square matrices, and matrices of column dimension 1 (column vectors) and row dimension 1 (row vectors). 

The Linear Algebraic Problem.—Familiarity with the basic theory of finite vectors and matrices—the notions of rank and linear dependence, the Cayley-Hamilton theorem, the Jordan normal form, orthogonality, and related principles—will be presupposed. In this section and the next, matrices will generally be represented by capital letters, column vectors by lower case English letters, scalars, except for indices and dimensions, by lower case Greek letters. The vectors a,b,x,y,..., will have elements au f it gt, r) . .. the matrices A, B,...,... 

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... 

In the general case we use the symbols U and V to represent projection matrices in 5" and S , each containing r projection vectors, and the symbols S and L to represent their images in the dual space ... 

Most of the algebra of vectors and matrices that is used in this chapter has been explained in Chapters 9 and 29. Small discrepancies between the tabulated values in the examples and their exact values may arise from rounding of intermediate results. 

From the latent vectors and singular values one can compute the nxr generalized score matrix S and the pxr generalized loading matrix L. These matrices contain the coordinates of the rows and columns in the space spanned by the latent vectors ... 

It should be appreciated that canonical correlation analysis, as the name implies, is about correlation not about variance. The first step in the algorithm is to move from the original data matrices X and Y, to their singular vectors, Ux and Uy, respectively. The singular values, or the variances of the PCs of X and Y, play no role. 

The score vector, zx, is found in a two-step operation to guarantee that the covariance of the scores is maximized. Once z, i, i, and qx have been found, the procedure is repeated for the residual matrices ExA and Ev i to find z2,ol2,u2, and q2. This continues until the residuals contain no... 

A note of caution is necessary when we let MATLAB generate the state space model from a transfer function. The vector C (from S. c) is [0 0.5], which means that the indexing is reversed such that x2 is the output variable, and xl is the derivative of x2. Secondly, C is not [0 1], and hence we have to rescale the matrices B and C. These two points are further covered in MALTA Session 4. 

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... 

Note that all functions in MATLAB, such as sqrt (), are smart enough that they accept scalars, vectors, and where appropriate, matrices.1... 

The principal topics in linear algebra involve systems of linear equations, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. 

Vectors and matrices are given as boldface symbols throughout.) E is the unit matrix G is a matrix which depends, only although not in a simple fashion, on the geometry and the atomic masses of the molecule. F is the matrix of force constants expressed in the 3 N — 6 independent internal coordinates. Its elements are ... 

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 ... 

In this chapter, we plan to use the information presented over the past three chapters to illustrate the geometry of vectors and matrices these concepts will continue to be discussed routinely throughout this series in relation to regression vectors. 

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