Eigenvalue analysis condition number

Certain quantitative measures from linear control theory may help at various steps to assess relationships between the controlled and manipulated variables. These include steady-state process gains, open-loop time constants, singular value decomposition, condition numbers, eigenvalue analysis for stability, etc. These techniques are described in... 

We have now identified when the linear system Ax = b will have exactly one solution, no solution, or an infinite number of solutions however, these conditions are rather abstract. Later, in our discussion of eigenvalue analysis, we see how to implement these conditions for specific matrices A and vectors b. 

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

The Smith-Ewart equations can be solved using a single numerical eigenvalue determination under all conditions. Analytical solutions can also be obtained if n is not too large (n< 0.7)(8,9,10). These solutions encompass both the steady state and the approach to the steady state. Thus the particle number concentrations N, N, . .are known once P, k and c have been determined experimentally. As will be seen, these populations are the starting point for the MWD analysis. 

Fig. 8. Representatives of the four conformations obtained in the M = 4 analysis and the conditional transition probabilities between them (lag time r = O.lps). Fat numbers indicating the statistical weight of each conformation, numbers in brackets the conditional probability to stay within a conformation. Flexibility in peptide angles is marked with arrows, cf. Fig. 7. Note that the transition matrix relating to this picture is not symmetric but reversible. Top left For the helix conformation the backbone is colored blue for illustrative purpose. It should be obvious from Fig. 5 that for significantly larger lag time r only two eigenvalues will correspond to metastability such that only the helical conformation and a mixed flexible and partially unfolded one remain with significantly high conditional probability to stay within...

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