The main result shows that under natural assumptions a matrix function that admits a thematic factorization also admits a monotone thematic factorization and the indices of a monotone thematic factorization are uniquely determined by the matrix function itself.
In a particular case when the matrix of linear terms is constant, a representation of the fundamental matrix given by a special matrix function (so-called delayed matrix exponential, etc)., for example, in [10, 11] (for a continuous case) and in [12, 13] (for a discrete case), can give concrete formulas expressing solution of the considered problem in analytical form.
(a), a subset of, is the state space of a continuous-time linear time-varying unforced dynamic system described for by (2.5) ., a subset of, is the state space of a continuous-time linear time-varying unforced dynamic system described for by (2.5). is a real matrix function of piecewise constant entries whose image is, where, for all with if.
On the other hand, by (1.16), (1.17), and (3.3) the matrix function A and constant matrices G l ( l = 1, …, m 0 ) satisfy, respectively, the inequalities (1.18) and (1.19).
We describe Wiener Hopf factorization indices of U in terms of superoptimal singular values of Φ and thematic indices of Φ−F, where F is a superoptimal approximation of Φ by bounded analytic matrix functions.
Moreover, these components actually are disjoint with the subgroup generated by the triangularizable matrix functions.
Such factorizations were introduced by V. V. Peller and N. J. Young (1994, J. Funct. Anal.120, 300 343) for studying superoptimal approximation by bounded analytic matrix functions.
To simplify the subsequent notation define the matrix function by Now, taking into account (3.6), define the nonnegative real functional by (3.7).
As a first methodological contribution, we prove that AIO can be uniquely described and computed, even in the case when there are metabolic system cycles, by a matrix whose coefficients are nonlinear functions of the flux variables.
A Jacobi-type parameter is determined in the pre-conditioner matrix by minimizing the Frobenius norm of the matrix function representing the difference between the IB forcing solution matrix and the pre-conditioner matrix.
