矩阵方程AX-XB=C的最小多项式解法

THE SOLVING PROCESS OF MATRIX EQUATION AX-XB=C BY MINIMAL POLYNOMIAL

关于矩阵方程AX—XB=C的解法有不少的论文,大部分是采用矩阵的拉直运算或拉直运算的变形方法求解,文献[1]给出了连分式解法,本文利用矩阵A,B的最小多项式求解此方程,使得方程的解比目前已见的结果较简洁,同时当B=-A^T稳定、C为任意正定矩阵时所构造的正定二次型Liapunov函数的表达式较目前的结果更明确、简单.

This paper surveys the solution of matrix equation AX - XB = C by means of theminimal polynomial of matrix A and B. And give a simple formula for solving the equation.The formula is a multinomial of matrix A, B and C, so that it simplifies greatly digitalcomputation. On the other hand, if B = -A^T is stable and C is a positive definite matrix,the constructed positive definite quadratic form Liapunov function will be clear.