Matrix Mathematics

We will use the symbolism of placing either brackets ([A]) around or a double underline ( 4 ) under terms that are matrices. A single underline ( x ) will be used to indicate a vector, i.e., a matrix with only one column. This should help us keep track of which quantities are matrices, which are vectors, and which are scalar terms. 

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The authors describe the use of a Taylor expansion to negate the second and the higher order terms under specific mathematical conditions in order to make any function (i.e., our regression model) first-order (or linear). They introduce the use of the Jacobian matrix for solving nonlinear regression problems and describe the matrix mathematics in some detail (pp. 178-181). 

In fact, it is true that equation 69-10 represents the least-squares solution to the problem of finding the coefficients of equation 69-3, it is just not obvious from this derivation, based on matrix mathematics. To demonstrate that equation 69-10 is, in fact, a least-squares solution, we have to go back to the initial problem and apply the methods of calculus to the problem. This derivation has been done in great detail [7], and in somewhat lesser detail in a spectroscopic context [8],... 

In this chapter, we present the basic matrix mathematics that is required for understanding the methods introduced later in the book. In line with the philosophy that all concepts are immediately implemented in Matlab and/or Excel, this will be done here as well. This way, Chapter 2 not only revises the basic mathematics, it also serves as a very short introduction to the Matlab and Excel languages. It is not meant to be a manual on Matlab or Excel the reader will need to refer to more specialised texts and proper manuals. Several more advanced features of both languages are not covered at this introductory stage but will be explained as they emerge in later chapters. 

As mentioned before, this chapter has two goals, (a) to refresh some basic matrix mathematics and (b) to familiarise the reader with the essentials of both Matlab and Excel, particularly with respect to multivariate data... 

In Chap. 15 we reviewed a tittle matrix mathematics and notation. Now that the tools are available, we will apply them in this chapter to the analysis of multivariable processes. Our primary concern is with closedloop systems. Given a process with its matrix of openloop transfer functions, we want to be able to see the effects of using various feedback controllers. Therefore we must be able to find out if the entire closedloop multivariable system is stable. And if it is stable, we want to know how stable it is. The last question considers the robustness of the controller, i.e., the tolerance of the controller to changes in parameters. If the system becomes unstable for small changes in process gains, time constants, or deadtimes, the controller is not robust. 

Excel solves systems of linear equations with a single statement. If the following matrix mathematics is unfamiliar to you, disregard it. The important result is the template in Figure 19-5 for solving simultaneous equations. You can use this template by following the instructions in the last paragraph of this section, even if the math is not familiar. 

Despite his skepticism, Pauli learned the intricacies of matrix mathematics and applied the Heisenberg version of the new quantum mechanics to the hydrogen atom. In less than three weeks, Pauli obtained the same formula that Bohr had obtained in 1913, only this time the route to the formula was a coherent theory—the new theory of quantum mechanics. Herewith, wrote Pauli, it has been demonstrated that the Balmer terms come out directly from the new quantum mechanics. So momentous was this demonstration that the skeptic Pauli became a believer in the matrix mathematical formulation of Born, Heisenberg, and Jordan. 

With his quantum mechanics, Schrodinger created a powerful formalism for treating atoms and molecules. Schrodinger s version of quantum mechanics was preferred by physicists through the 1930s. As the years passed, however, physicists mastered the matrix mathematics that was the basis of Heisenberg s quantum mechanics and with that mastery they discovered that some problems are treated more naturally with the matrix approach. Today, both approaches are used equally. 

Matrix.xls illustrates the tools available for matrix mathematics, with examples. Polar.xls illustrates how to convert from polar to Cartesian coordinates. 

Matrix-type delivery systems are simple to make release is usually controlled by diffusion of drug through the polymer matrix. Mathematical descriptions of release are more complicated than are obtained for membrane-type devices, and it is difficult to produce devices that provide a constant rate of release. However, these materials are versatile and almost any compound can be formulated into a controlled-release matrix. 

Matrix Mathematical structure for arranging numbers or symbols that have particular mathematical rules for use. 

The condition number analysis is used after the RGA analysis to check the ease of the control structure. The condition number is commonly used as an index of controllability which is a ratio of the largest singular value to the smallest nonzero singular value of a matrix, i.e. the HEN gain matrix. Mathematically, condition number can be defined as a positive number that provides useful information on the sensitivity of matrix properties to variations in the elements of the matrix. In terms of process control, a large condition number indicates that it will be impractical, if not impossible, to satisfy the entire set of control objectives and vice versa for small values. Therefore, it is used to measure the performance of the structured control system by RGA of the designed HEN. This condition number analysis is used mainly... 

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