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Noncommutative Computer Algebra in Linear Algebra and Control Theory


Frank Dell Kronewitter

We will show how noncommutative computer algebra can be quite useful in solving problems in linear algebra and linear control theory. Such problems are highly noncommutative, since they typically involve block matrices. Conventional (commutative) computer algebra systems cannot handle such problems and the noncommutative computer algebra algorithms are not yet well understood. Indeed, the Grobner basis algorithm, which plays a central role in many computer algebra computations, is only about thirteen years old in the noncommutative case.

We will demonstrate the e ectiveness of our algorithms by investigating the partially prescribed matrix inverse completion problem and computations involv- ing singularly perturbed dynamic systems. On both of these sorts of problems our methods proved to be quite e ective. Our investigations into the partially prescribed matrix completion problem resulted in formulas which solve all 3x3 problems with eleven known and seven unknown blocks. One might even say that these formulas represent 31,824 new theorems. Our singular perturbation efforts focus on both the standard singular perturbed dynamic system and the informa- tion state equation. Our methods easily perform the sort of calculations needed to find solutions for the standard singular perturbation problem and in fact we are able to carry the standard expansion out one term further than has been done previously. We are also able to generate (new) formulas for the solution to the singularly perturbed information state equation.

After demonstrating how useful our methods are, we will pursue the formal analysis of our techniques which are generally refered to as Strategies. The formal definition of a strategy allows some human intervention and therefore is not as rigid as an algorithm. Still, a surprising amount of rigorous analysis can be done especially when one adds some simple hypotheses as we will. In particular, we will introduce the notion of a good polynomial and the gap of a polynomial ideal which will prove useful in our formal analysis. We will show how successful strategies correspond to low gap ideals. Also introduced is the strategy+, which allows the user a bit more freedom than a strategy.

The lion's share of our noncommutative computer algebra investigations have been in the field of linear system theory. We will describe our accomplishments and demonstrate the strategy technique with some highly algebraic theorems on positive real transfer functions.

Finally, we will turn to controllability and observability operators which play an important role in linear system theory and are expressed symbolically as in nite sequences. We will offer finite algebraic characterizations of these operators, and use these characterizations to derive the state space isomorphism theorem.

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