We analyze different approaches to differential-algebraic equations with attention to the implemented rank conditions of various matrix functions. These conditions are apparently very different and certain rank drops in some matrix functions actually indicate a critical solution behavior. We look for common ground by considering various index and regularity notions from literature generalizing the Kronecker index of regular matrix pencils. In detail, starting from the most transparent reduction framework, we work out a comprehensive regularity concept with canonical characteristic values applicable across all frameworks and prove the equivalence of thirteen distinct definitions of regularity. This makes it possible to use the findings of all these concepts together. Additionally, we show why not only the index but also these canonical characteristic values are crucial to describe the properties of the DAE.

翻译：我们分析了处理微分代数方程的不同方法，重点关注各种矩阵函数所实现的秩条件。这些条件表面上差异显著，且某些矩阵函数中的特定秩降实际上指示了临界解行为。通过考虑文献中推广正则矩阵束Kronecker指标的各种指标与正则性概念，我们寻求其共同基础。具体而言，从最透明的约简框架出发，我们构建了一个适用于所有框架的、具有规范特征值的综合性正则性概念，并证明了十三种不同正则性定义的等价性。这使得综合运用所有这些概念的研究成果成为可能。此外，我们阐明了为何不仅指标值，这些规范特征值对于描述DAE的特性也至关重要。