An Introduction to Matrix Algebra

Matrix algebra provides a powerful method for the manipulation of sets of numbers. Many mathematical operations — addition, subtraction, multiplication, division, etc. — have their counterparts in matrix algebra. Our discussion will be Umited to the manipulations of square matrices. For purposes of illustration, two 3x3 matrices will be defined, namely 

The following examples illustrate addition, subtraction, multiplication and division using a constant. 

Addition or subtraction of two matrices (both must contain the same number of rows and columns)  

Multiplication of two matrices can be either scalar multiplication or vector multiplication. Scalar multiplication of two matrices consists of multiplying corresponding elements, i.e.. 

Thus it s clear that both matrices must have the same dimensions m x n. Scalar multiplication is commutative, that is, A B = B A 

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

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.)... 

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. 

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. 

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... 

Various methods have been developed for dealing with the anomalous commutation relationships in molecular quantum mechanics, chief among them being Van Vleck s reversed angular momentum method [10]. Most of these methods are rather complicated and require the introduction of an array of new symbols. Brown and Howard [15], however, have pointed out that it is quite possible to handle these difficulties within the standard framework of spherical tensor algebra. If matrix elements are evaluated directly in laboratory-fixed coordinates and components are referred to axes mounted on the molecule only when necessary, it is possible to avoid the anomalous commutation relationships completely. Only the standard equations given earlier in this chapter are used to derive the required results it is just necessary to keep a cool head in the process ... 

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