An iterative method for the symmetric and skew symmetric solutions of a linear matrix equation AXB+CYD=E
An iterative method for the symmetric and skew symmetric solutions of a linear matrix equation AXB+CYD=E
复制标题
DOI：
10.1016/j.cam.2009.11.052
复制
发表时间：
2010-04
期刊：
影响因子：
--
通讯作者：
Xingping Sheng;Guoliang Chen
中科院分区：
文献类型：
--
作者：
Xingping Sheng;Guoliang Chen
本文给出了求解线性矩阵方程AXB+CYD=E的对称解和反对称解的两种有效的迭代方法，分别用真实的对矩阵X和Y.通过这两种迭代方法，可以自动地判定矩阵方程的对称解和反对称解的可解性。当矩阵方程存在对称解和反对称解时，对任意初始对矩阵X0和Y0，在无舍入误差的情况下，可以在有限迭代步内得到对称解和反对称解，并且通过选择一类特殊的初始对矩阵，可以得到对称解和反对称解的最小范数.另外，通过求一个新的矩阵方程AX B+CY D=E的最小范数解，可以得到给定矩阵X和Y在Frobenius范数下的唯一最佳逼近对解X和Y，其中E D.数值算例表明，该迭代方法是有效的。
In this paper, two efficient iterative methods are presented to solve the symmetric and skew symmetric solutions of a linear matrix equation AXB+CYD=E, respectively, with real pair matrices X and Y. By these two iterative methods, the solvability of the symmetric and skew symmetric solutions for the matrix equation can be determined automatically. When the matrix equation has symmetric and skew symmetric solutions, then, for any initial pair matrices X0and Y0, symmetric and skew symmetric solutions can be obtained within finite iteration steps in the absence of roundoff errors, and the minimum norm of the symmetric and skew symmetric solutions can be obtained by choosing a special kind of initial pair matrices. In addition, the unique optimal approximation pair solution X̂ and Ŷ to the given matrices X¯ and Y¯ in Frobenius norm can be obtained by finding the minimum norm solution of a new matrix equation AX˜B+CY˜D=E˜, where E˜=E−AX¯B−CY¯D. The given numerical examples demonstrate that the iterative methods are quite efficient.