We present here a new splitting method to solve Lyapunov equations in a Kronecker product form. Although this resulting matrix is of order $n^2$, each iteration demands two operations with the matrix $A$: a multiplication of the form $(A-\sigma I) \tilde{B}$ and a inversion of the form $(A-\sigma I)^{-1}\tilde{B}$. We see that for some choice of a parameter the iteration matrix is such that all their eigenvalues are in absolute value less than 1. Moreover we present a theorem that enables us to get a good starting vector for the method.

翻译：本文提出了一种新的分裂方法来求解Kronecker积形式的Lyapunov方程组。尽管最终得到的矩阵阶数为$n^2$，但每次迭代仅需对矩阵$A$进行两次运算：一次形如$(A-\sigma I) \tilde{B}$的乘法运算和一次形如$(A-\sigma I)^{-1}\tilde{B}$的求逆运算。我们发现在特定参数选择下，迭代矩阵的所有特征值的绝对值均小于1。此外，我们提出一个定理，使得该方法能够获得良好的初始向量。