Abstract

We execute the complete integration of the simplest matrix Riccati equation in the two- and three-dimensional cases for an arbitrary linear differential operator. The solution is constructed in terms of the Jordan form of an unknown matrix and the corresponding similarity matrix. We show that a similarity matrix is always representable as the product of two matrices one of which is an invariant of the differential operator.

中文翻译：

---

关于矩阵Riccati方程的积分

摘要

对于任意线性微分算子，我们在二维和三维情况下执行最简单矩阵Riccati方程的完全积分。根据未知矩阵的约旦形式和相应的相似性矩阵构造解决方案。我们证明相似矩阵总是可以表示为两个矩阵的乘积，其中两个是微分算子的不变式。