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矩阵方程 的自反和反自反矩阵解
摘要:如果 满足条件:(1) ,(2) ,则称 为广义反射矩阵,广义反射矩阵也是自伴的对合矩阵。设 和 都是广义反射矩阵,如果 满足 ,则称 为关于矩阵对 的广义(反)自反矩阵;如果 满足 ,则 称为关于矩阵 的广义(反)自反矩阵。这篇介绍了矩阵方程 ,在系数矩阵 , 为广义(反)自反矩阵的条件下,(反)自反矩阵解存在的充分必要条件及表达形式。另外,研究了矩阵方程 的(反)自反矩阵解集 ,利用矩阵的分解,导出(反)自反矩阵问题的最佳逼近解。
关键词:自反矩阵;反自反矩阵;矩阵方程;Frobenius范数;矩阵最佳逼近问题
The reflexive and anti-reflexive solutions of the
matrix equation
Abstract :An complex matrix is said to be a generalized reflection matrix if and .An complex matrix ia said to be a reflexive (or anti-reflexive) matrix with respect to the generalized reflection matrixs , if . An complex matrix ia said to be a reflexive (or anti-reflexive) matrix with respect to the generalized reflection matrix , if .This paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti-reflexive with respect to a generalized reflection matrixs solutions of the matrix equation .In addition, incorresponding solution set of the equation.The explicit expression of the nearest matrix to a given matrix in the Frobenius noum have been provided.
Keywords:Reflexive matrix; Anti-reflexive matrix; Matrix equation; Frobenius norm; Matrix nearness problem.