Introduction to Matrix Algebra

In Chapter 2, we give a brief introduction to matrix algebra and its implementation in Matlab and Excel. 

Basic understanding and efficient use of multivariate data analysis methods require some familiarity with matrix notation. The user of such methods, however, needs only elementary experience it is for instance not necessary to know computational details about matrix inversion or eigenvector calculation but the prerequisites and the meaning of such procedures should be evident. Important is a good understanding of matrix multiplication. A very short summary of basic matrix operations is presented in this section. Introductions to matrix algebra have been published elsewhere (Healy 2000 Manly 2000 Searle 2006). 

Edwards, A. L. 1984. An Introduction to Linear Regression and Correlation, 2nd edn, W. H. Freeman, New York. (Clearly written treatment, with a good introduction to matrix algebra.)... 

An introduction to matrix algebra will be found in Appendix V. 

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. 

Hochschild, G. Introduction to Affine Algebraic Groups (San Francisco Holden-Day, 1971). Mainly algebraic matrix groups, with Hopf-algebraic treatment. The emphasis is on characteristic zero and relation with Lie algebras. 

This book is an introduction to computational chemistr y, molecular mechanics, and molecular orbital calculations, using a personal mieroeomputer. No speeial eom-putational skills are assumed of the reader aside from the ability to read and write a simple program in BASIC. No mathematieal training beyond ealeulus is assumed. A few elements of matrix algebra are introdueed in Chapter 3 and used throughout. 

Sections on matrix algebra, analytic geometry, experimental design, instrument and system calibration, noise, derivatives and their use in data analysis, linearity and nonlinearity are described. Collaborative laboratory studies, using ANOVA, testing for systematic error, ranking tests for collaborative studies, and efficient comparison of two analytical methods are included. Discussion on topics such as the limitations in analytical accuracy and brief introductions to the statistics of spectral searches and the chemometrics of imaging spectroscopy are included. 

The well-known GF matrix technique of E. B. Wilson and his colleagues for calculating the harmonic frequencies of polyatomic molecules is based on the use of valence coordinates, also referred to as internal coordinates. What is presented here is merely a sketch of the method a fuller discussion would require extensive use of matrix algebra, which is beyond the scope of this book. The appendix on matrices in this chapter serves only as a very short introduction to such methods. For details reference should be made to the classical work of E. B. Wilson, J. C. Decius and P. C. Cross (WDC) in the reading list. 

Following the very brief introduction to the method of lines and differential-algebraic equations, we return to solving the boundary-layer problem for nonreacting flow in a channel (Section 7.4). From the DAE-form discretization illustrated in Fig. 7.4, there are several important things to note. The residual vector F is structured as a two-dimensional matrix (e.g., Fuj represents the residual of the momentum equation at mesh point j). This organizational structure helps with the eventual software implementation. In the Fuj residual note that there are two timelike derivatives, u and p (the prime indicates the timelike z derivative). As anticipated from the earlier discussion, all the boundary conditions are handled as constraints and one is implicit. That is, the Fpj residual does not involve p itself. 

The matrix-algebraic representation (9.20a-e) of Euclidean geometrical relationships has both conceptual and notational drawbacks. On the conceptual side, the introduction of an arbitrary Cartesian axis system (or alternatively, of an arbitrarily chosen set of basis vectors ) to provide vector representations v of geometric points V seems to detract from the intrinsic geometrical properties of the points themselves. On the notational side, typographical resources are strained by the need to carefully distinguish various types of... 

Although matrix multiplications, row reductions, and calculation of null spaces can be done by hand for small matrices, a computer with programs for linear algebra are needed for large matrices. Mathematica is very convenient for this purpose. More information about the operations of linear algebra can be obtained from textbooks (Strang, 1988), but this section provides a brief introduction to making calculations with Mathematica (Wolfram, 1999). 

This text also provides a general introduction to vector and matrix -algebraic methods that are highly effective for thermodynamics as well as applications in quantum mechanics and other advanced topics in physical chemistry. 

Matrix algebra is a key mathematical tool in doing modern-day quantum-mechanical calculations on molecules. Matrices also furnish a convenient way to formulate mudi of the theory of quantum mechanics. TTiis section therefore gives an introduction to matrices. Matrix methods will be used in later chapters, but this book is written so that the material on matrices can be omitted if time does not allow this material to be covered. 

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