Matrix algebra exponential

This section is not a substitute for one of the many good texts on mathematical methods written for scientists with different backgrounds. No one of these volumes will appeal to everybody, but I find Boas (1966) has the dearest and most comprehensive coverage of the mathematical problems arising in the present volmne. It is intended that the brief summary of matrix algebra will help the reader to follow those sections of the book in which kinetic equations are derived. Specific examples of the derivation of rate equations by this method, including munerical evaluation of exponential coefficients and amplitudes, are foimd in sections 4.2 and 5.1. 

The proof of this proposition follows fairly easily from the definition of matrix exponentiation and standard techniques of vector calculus. See any linear algebra textbook, such as [La, Chapter 9]. 

First order series/parallel chemical reactions and process control models are usually represented by a linear system of coupled ordinary differential equations (ODEs). Single first order equations can be integrated by classical methods (Rice and Do, 1995). However, solving more than two coupled ODEs by hand is difficult and often involves tedious algebra. In this chapter, we describe how one can arrive at the analytical solution for linear first order ODEs using Maple, the matrix exponential, and Laplace transformations. 

Higher order ODEs (of order n) were converted to a system of n coupled linear first order ODEs in section 2.1.4. This system was then solved using the exponential matrix developed earlier. This approach yields analytical solutions for linear ODEs of any order. In section 2.1.5, the given system of coupled linear ODEs was converted to Laplace domain. The resulting linear system of algebraic equations was then solved for the solution in the Laplace domain. The solution obtained in the Laplace domain was then converted to the time domain. 

The first part of the programme consists of the calculation of the % matrix elements which form the coefficients of the system of equations. % The second part of the programme, as this has been explained in section % 2.1.2, consists of the iterative application of the L Hospital s rule % for the calculation of the solution of these equations that make up the % coefficients of the family of new tenth algebraic order exponentially % fitted methods for the numerical integration of the Schrodinger type % equations. 

The explicit form of the matrix representation of the operator Rx(0p ) can be obtained with some algebraic manipulation of the properties of exponential operators in the simple... 

The solution of the set of linear ordinary differential equations is very cumbersome to evaluate in the form of Eq. (5.40), because it requires the evaluation of the inOnite series of the exponential term e. However, this solution can be modified by further algebraic manipulation to express it in terms of the eigenvalues and eigenvectors of the matrix A. 

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