Abstract The notions of semilalgebra and its associated subalgebra are introduced. Linear spaces of row-column, skew-angle, and acute-angle matrices that are subalgebras or semialgebras of the full matrix algebra are also defined. Conditions for the splittability and solvability by quadratures of a matrix linear ordinary differential equation of the first order with two-sided multiplication $$dX/dt=\sum _{k=1}^K A_k X B_k+C$$ are studied, where $$K\in \mathbb {N} $$ and $$X $$ and $$A_k $$ , $$B_k$$ , $$C $$ are, respectively, the unknown and given matrix-valued continuously differentiable functions of the variable $$t\in \mathbb {R} $$ ranging in one of these linear spaces.

中文翻译：

---

特殊矩阵上的一阶线性常微分方程

摘要 介绍了半代数及其相关子代数的概念。还定义了作为全矩阵代数的子代数或半代数的行-列、斜角和锐角矩阵的线性空间。研究了具有双边乘法的一阶矩阵线性常微分方程的可分裂性和可解性条件 $$dX/dt=\sum _{k=1}^K A_k X B_k+C$$，其中 $$K\in \mathbb {N} $$ 和 $$X $$ 和 $$A_k $$ , $$B_k$$ , $$C $$ 分别是未知的和给定的矩阵值连续可微变量 $$t\in \mathbb {R} $$ 在这些线性空间之一中的函数。