Algebra and Matrices

When we glance across the main subjects areas, methods, algorithms and codes of this book, we see the following key elements  

We have encountered several scalar equations, some of which are transcendental (Chapter 3), some involve matrices (Chapter 6) but most equations and models are differential in nature (Chapters 4, 5, 6, 7), involving time or location dependent functions for dynamic models of various chemical and biological engineering plants and apparatus. 

A first course on DEs typically studies how to solve equations of the form y (t) = f(t,y(t)). Such a course develops methods to find explicit solutions for specific (theoretically solvable) classes of differential equations and besides, it studies the behavior of solutions of DEs for which there are or are not any known explicit solution methods. In a nutshell, such a course looks at DEs and their solutions both quantitatively and qualitatively. 

Unfortunately, the number of classes of DEs with known explicit solutions is rather small. Worse, these theoretically solvable DEs do not often appear, if at all, in practical models such as are needed by the chemical and biological engineer, except for linear DEs which we will treat later in this appendix. 

How does one handle DEs that have no known explicit algebraic nor formulistic solution such as the ones of this book How can they even be solved  

---

Due to the special structure of MATLAB, readers should be familiar with the mathematical concepts pertaining to matrices, such as systems of linear equations, Gaussian elimination, size and rank of a matrix, matrix eigenvalues, basis change in n-dimensional space, matrix transpose, etc. For those who need a refresher on these topics there is a concise Appendix on linear algebra and matrices at the end of the book. 

Why do we need to understand Linear Algebra and Matrices for solving DEs. 

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,...,... 

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. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

It is the objective of the present chapter to define matrices and their algebra - and finally to illustrate their direct relationship to certain operators. The operators in question are those which form the basis of the subject of quantum mechanics, as well as those employed in the application of group theory to the analysis of molecular vibrations and the structure of crystals. 

The algebra of matrices gives rules for (1) equality, (2) addition and subtraction, (3) multiplication, and (4) division as well as (5) an associative and a distributive law. It also includes definitions of (6) a transpose, adjoint and inverse of a matrix. 

These challenges can be dealt with the powerful mathematical tools of quantum chemistry, as advocated by Per-Olov Lowdin.[l, 2, 3, 4] In our studies, linear algebras with matrices,[4] partitioning techniques,[3] operators and superoperators in Liouville space, and the Liouville-von Neumann... 

APPLIED ANALYSIS, Cornelius Lanczos. Classic work on analysis and design ol linite processes for approximating solution of analytical problems. Algebraic equations, matrices, harmonic analysis, quadrature methods, much more. 559pp. 5H x 8H. 65656-X Pa. 11.95... 

In quantum mechanics we often encounter associative algebras of operators and matrices which are noncommutative. For example, the set of all n x n matrices over the real or complex number fields is an n2-dimensional vector space which is also an associative, noncommutative algebra whose multiplication is just the usual matrix multiplication. Also, the subset of all diagonal n x n matrices is a commutative algebra. 

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. 

More details on matrices and their manipulation are available in Appendix 1 of F. A. Cotton, Chemical Applications of Group Theory, 3rd ed., John Wiley Sons, New York, 1990, and in linear algebra and finite mathematics textbooks. 

In general there are two ways to solve Eq. (L.3) for Xi,. . ., x elimination techniques and iterative techniques. Both are easily executed by computer programs. In the pocket in the back cover of this book you will find a disk containing Fortran computer programs that can be used in solving sets of linear equations. We shall illustrate the Gauss-Jordan eliinination method. Other techniques can be found in texts on matrices, linear algebra, and numerical analysis. 

The rectangular array of cells is labeled A, B, C, D... across the top, and 1, 2, 3, 4,. .. down the left-hand side (Fig. A.l). Cell B3 is in the second column (B column) and the third row from the top. (As in linear algebra with matrices, columns go down and rows go across.)... 

For the lecture of the present work, the readers can refer directly to Sect. 3.1.2 below, then to the real formalism of the Clifford algebras and avoid all that concerns the correspondence between this formalism and the one of the complex matrices and spinors. However, the knowlewge of this correspondence is recommended. 

The matrix product is well known and can be found in any linear algebra textbook. It reduces to the vector product when a vector is considered as an I x 1 matrix, and a transposed vector is a 1 x I matrix. The product ab is the inner product of two vectors. The product ab is called the outer product or a dyad. See Figure 2.2. These products have no special symbol. Just putting two vectors or matrices together means that the product is taken. The same also goes for products of vectors with scalars and matrices with scalars. 

---