几种特殊循环矩阵的PROCRUSTES问题  

PROCRUSTES PROBLEMS FOR SEVERAL SPECIAL CIRCULANT MATRICES

作　　者：邓远北[1] 刘莹[1] 杨娟[1] 

机构地区：[1]湖南大学数学与计量经济学院,长沙410082

出　　处：《数值计算与计算机应用》2016年第1期11-24,共14页Journal on Numerical Methods and Computer Applications

摘　　要：本文研究了在Hankel—循环矩阵和Hankel—反循环矩阵的约束下矩阵方程组AX=B,XC=D的最小二乘解问题.结合最优化理论和循环矩阵的性质,将其转化为简单的线性方程Qy=b的求解问题,得到了通解的表达式.进一步,证得系数矩阵Q是一个与所求矩阵X相关联的循环矩阵,从而找到了解唯一的充分必要条件并给出了解的表达式.此外,借助于矩阵的广义1—范数,给出了有唯一解的判定条件.最后,给出了具体的算法和算例.The least-squares problems of the matrix equations AX = B and XC = D for several special circulant matrices are discussed in this paper. Combining the optimization theory and the properties of circulant matrices, the least-squares problem can be transformed into a simple problem of linear equation Qy = b. Therefore, the explicit representations of the general solution can be derived. Moreover, the coefficient matrix Q of the linear equation related to the matrix X with constraint conditions is a circulant matrix. Thus, the explicit representations of the general and unique solution are derived. Besides, the determination conditions of the unique solution are obtained by the generalized 1 norm. Finally, the algorithm and example are given.

关 键 词：矩阵方程 最小二乘问题 PROCRUSTES问题 Hankel--循环矩阵 Hankel--反循环矩阵. 

分 类 号：O151.21[理学—数学]