Eigenvalue theory

Let us define matrix P as a product of series impedance matrix Z and shunt admittance matrix Y for a multiconductor system  

Applying the eigenvalue theory, off-diagonal matrix P can be diagonalized by the following matrix operation  

The notation of matrix [ ] and vector () is, hereafter, omitted for simplification. Rewriting the earlier equation. 

Since Q is the diagonal matrix, only the kth column of A is multiplied by the kth diagonal entry of Q when calculating AQ. Therefore, the following equation is satisfied for each k  

The following equation is obtained for the fcth column by substituting the earlier equation into Equation 1.126  

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A useful tool to enhance the understanding of eigenvalue theory is an eigenvalue map [9], which shows discreet regions of similar node behavior at a set reflux. An example is depicted in Figure 3.23. 

The approach to calculation of the minimum reflux mode, based on eigenvalue theory, was introduced in the work (Poellmann, Glanz, Blass, 1994). In contrast to the above-mentioned works of Doherty and his collaborators this method calculates the mode of minimum reflux not only for direct and indirect, but also for intermediate split of four-component mixtures. 

Poellmann, R, Glanz, S., Blass, E. (1994). Calculating Minimum Reflux of Nonideal Multicomponent Distillation Using Eigenvalue Theory. Comput. Chem. Eng., 18,549-53. 

As discussed in Section 1.4.1, analysis of a multiconductor system requires a number of computations of functions. The application of eigenvalue theory makes it easy to calculate matrix functions. This is a major advantage of eigenvalue theory. 

One way to calculate matrix functions without eigenvalue theory is to use series expansions. The following series expansions are often used to calculate matrix functions ... 

Using eigenvalue theory, a matrix function is given by... 

Note that the computation of Equation 1.112 is made possible only by using eigenvalue theory, as in Equation 1.138. 

In this section, we have discussed the method that directiy applies eigenvalue theory. However, it is not efficient in terms of numerical computations as it requires the product of off-diagonal matrices. The method will be more complete with modal theory. 

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