矩阵方程AXAT=C的对称斜反对称解

Solutions of Matrix Equation AXAT = C in Symmetric and Skew-Antisymmetric Matrix Set

中文关键词:  矩阵方程  对称斜反对称矩阵  广义奇异值分解  极小范数解

英文关键词:matrix equation  symmetric and skew-antisymmetric matrix  generalized singular-value decomposition  minimal norm solution.

中文摘要:

      设A∈R^m×n,C∈R^m×m给定,利用矩阵的广义奇异值分解和对称斜反对称矩阵的性质,得到了矩阵方程(1)AXA^T=C存在对称斜反对称解的充要条件和通解表达式;证明了若方程(1)有解,则一定存在唯一极小范数解,并给出了极小范数解的具体表达式和求解步骤.

英文摘要:

      By applying the generalized singular-value decomposition (GSVD) of matrix pairs and the properties of symmetric and skew-antisymmetric , we obtain the sufficient and necessary conditions under which the matrix equation AXA^T = C is solvable in n × n symmetric and skew-antisymmetric matrix set, and prove that if the above equation is solvable, then there exists a unique minimal norm solution, and give the procedure to find this solution, where A ∈ R^m×n and C ∈ R^m×m are given matrices.