Jordan decomposition

II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part... 

The argument at the start of the proof shows now that any homomorphism G-> H preserves Jordan decompositions. In particular, Jordan decomposition in an algebraic matrix group is intrinsic, independent of the choice of an embedding in GL . 

Additive Jordan Decomposition) Let it be a perfect field, Tannxn matrix. Show there are unique R and S with R nilpotent, S separable, RS = SR, and R + S=T. 

Give an example to show that the Jordan decomposition need not exist over a field that is not perfect. 

Since the last reaction is a linear combination of the first two (sum), it can be easily proved that the rank remains unchanged at 2. So to conclude, the concentrations of all components in this network can be expressed in terms of two, say H2 and Freon 12, and the first two reactions form an independent reaction set. In case of more complicated networks it may be difficult to determine the independent reactions by observation alone. In this case the Gauss-Jordan decomposition leads to a set of independent reactions (see, e.g., Amundson, Mathematical Methods in Chemical Engineering—Matrices and Their Application, Prentice-Hall International, New York, 1966). 

The recent trend is that very large reaction mechanisms are created either manually or automatically for the combustion or the atmospheric decomposition of large organic molecules [see e.g. (Herbinet et al. 2010 Westbrook et al. 2011)], as discussed in Sect. 3.1. In such mechanisms, only a minority of the reaction steps has an experimentally measured rate coefficient, and most of the reaction parameters are estimated using simple rules. Therefore, it is common that many rate coefficients are identical within such mechanisms. A numerical consequence can be that an eigenvalue — eigenvector decomposition of the Jacobian does not exist. However, even in this case, the effect of concentration perturbations can always be studied on the basis of the Jordan decomposition of the Jacobian (Nagy and Turanyi 2009), as discussed below. 

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