We investigate the main phase space properties of the QR-flow when restricted to upper Hessenberg matrices. A complete description of the linear behavior of the equilibrium matrices is given. The main result classifies the possible $ \alpha $- and $ \omega $-limits of the orbits for this system. Furthermore, we characterize the set of initial matrices for which there is convergence towards an equilibrium matrix. Several numerical examples show the different limit behavior of the orbits and illustrate the theory.

中文翻译：

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上Hessenberg实矩阵的QR流动力学

我们研究了仅限于上Hessenberg矩阵时QR流的主要相空间特性。给出了平衡矩阵线性行为的完整描述。主要结果对该系统的轨道可能的$ \ alpha $-和$ \ omega $-极限进行分类。此外，我们表征了初始矩阵的集合，对于这些集合，它们趋向于一个平衡矩阵。几个数值示例说明了轨道的不同极限行为，并说明了该理论。