In an isolated lossless system where s? in Eq. (1) is negligible, modes associated with λqλq follow exponential progressions in directions associated with aqaq and bqbq. Specifically, λq>0λq>0 correspond to growing modes, and λq<0λq<0 correspond to decaying modes. It is, however, noted that the direction of the eigenvectors are not unique, as they JNJ-26481585 can point either in forward or backward direction, i.e. they are determined up to a sign. A decomposition of the source term ω? into directions of chemical modes χ? follows from AΣχ?=ω?. Using the identity AΣ=B-1AΣ=B-1, chemical mode amplitudes are calculated asequation(4)χ?=Bω?i.e. effective reaction rates of chemical modes are obtained by projecting the original source terms onto the left eigenvectors. Accordingly, the eigenvectors obtained by the eigenvalue decomposition represent source modes.
The mode amplitudes χ? can be linked to chemical reactions by relating them to reaction rates as ω?=Sπ?, where SS is the stoichiometric coefficient matrix and π? is the vector of net reaction rates. It is noted that SS includes an additional row to account for heat release in ω?. Accordingly, Eq. (4) can be expressed asequation(5)χ?=BSπ?Here, the mapping can be interpreted as the projection of net reaction rates onto eigenvectors from a higher-dimensional, but rank-deficient space that describes chemical reactions. In anabolic reactions larger space, a Q-dimensional subspace describes a manifold of valid reaction modes.