Lyapunov方程的线性变换解法

Linear Transformation Approach for Lyapunov Equation

利用线性变换以及方阵的Ｊｏｒｄａｎ标准型的方法，给出了Ｌｙａｐｕｎｏｖ矩阵方程存在唯一解的充分必要条件以及解的形式，采用初等的方法，得出的结果比已有结论丰富。该方法也可用来研究一般城上的矩阵方程，从而为研究密码学基础理论提供一种新方法。

By using linear transformation method, this paper gives the necessary and sufficient conditions for Lyapunov equation. The unique solutions are obtained. The method in this paper is elementary and the results are richer than those in references. The method can also be used to deal with matrix equations on general field, which supplies a new method to cryptography.