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 effectiveness of our algorithms by investigating the partially prescribed matrix inverse completion problem and computations involving singularly perturbed dynamic systems. On both of these sorts of problems our methods proved to be quite effective. Our investigations into the partially prescribed matrix completion problem resulted in formulas which solve all 3 x 3 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 information 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 infinite 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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机译：我们将展示非交换计算机代数如何在解决线性代数和线性控制理论中的问题时非常有用。这样的问题是高度不可交换的，因为它们通常涉及块矩阵。传统的（可交换）计算机代数系统无法处理此类问题，并且非可交换计算机代数算法尚未得到很好的理解。确实，在许多计算机代数计算中起着核心作用的Grobner基算法在非可交换情况下只有13年的历史。我们将通过研究部分规定的矩阵逆完成问题和涉及以下内容的计算来证明我们算法的有效性奇摄动的动力系统。在这两种问题上，我们的方法都被证明是非常有效的。我们对部分规定的矩阵完成问题的研究得出了可以用11个已知块和7个未知块解决所有3 x 3问题的公式。甚至可以说这些公式代表了31,824个新定理。我们对奇异摄动的研究集中于标准奇异摄动系统和信息状态方程。我们的方法可以轻松地执行所需的计算，以找到标准奇异摄动问题的解，并且实际上，我们能够比以前更进一步地扩展标准项。我们还能够生成（新的）公式来求解奇摄动的信息状态方程。在说明了我们的方法的实用性之后，我们将对我们的技术进行形式化分析，这些方法通常被称为“策略”。策略的正式定义允许某些人为干预，因此不像算法那样严格。仍然可以进行大量令人惊讶的严格分析，尤其是当我们添加一些简单的假设时。特别是，我们将介绍良好多项式的概念和理想多项式的差距，这将在我们的形式分析中被证明是有用的。我们将展示成功的策略如何与低差距理想相对应。还介绍了strategy +，它给用户带来了比策略更多的自由。;在我们的非交换计算机代数研究中，绝大部分都属于线性系统理论领域。我们将描述我们的成就并通过关于正实传递函数的一些高度代数定理来证明该策略技术。最后，我们将转向在线性系统理论中起着重要作用并象征性地表示为无限序列的可控制性和可观察性算子。我们将提供这些算子的有限代数表征，并使用这些表征导出状态空间同构定理。
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