Review of scalar, vector, and matrix operations

Our discussion is intended only to provide a foundation in linear algebra for the practice of numerical computing, and is continued in Chapter 3 with a discussion of matrix eigenvalue analysis. For a broader, more detailed, study of linear algebra, consult Strang (2003) or Golnb van Loan (1996). 

As we nse vector notation in our discussion of linear systems, a basic review of the concepts of vectors and malrices is necessary. 

Most often in basic mathematics, we work with scalars, i.e., single-valued numbers. These may be real, such as 3,1.4,5/7,3.14159, or they may be complex, 1 + 2i, l/2i, where 

Note that the product zz is always real and nonnegative, 

Using the important Au/gr formula, a proof of which is found in the supplemental material found at the website that accompanies this book. 

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It may be worth while to review the different kinds of multiplicity involved in the symbols appearing in Eqs. (3-6) and (3-15). Equation (3-6) is merely a shorthand way of writing the material balance for each of the key components, each term being a row matrix having as many elements as there are independent reactions. The equation asserts that when these matrices are combined as indicated, each element in the resulting matrix will be zero. The elements in the first two terms are obtained by vector differential operation, but the elements are scalars. Equation (3-15), on the other hand, is a scalar equation, from the point of view of both vector analysis and matrix algebra, although some of its terms involve vector operations and matrix products. No account need be taken of the interrelation of the vectors and matrices in these equations, but the order of vector differential operators and their operands as well as of all matrix products must be observed. 

Energy, which is the observable quantity associated with the Hamiltonian operator, is a pure number or, more precisely, a scalar quantity. Therefore, the Hamiltonian must be a scalar operator. In this section, the prescriptions for constructing a scalar operator from combinations of more complicated operators, such as vector angular momenta, and for evaluating matrix elements of these composite scalar operators are reviewed briefly. 

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