一类矩阵方程的中心对称定秩解及其最佳逼近

Central- Symmetric Minimal Rank Solutions and Its Optimal Approximation to A Class of Matrix Equation

通过采用一种新方法得出了矩阵方程AXB=C有中心对称解的充分必要条件、解的一般表达式;利用矩阵对的商奇异值分解、广义逆,给出了其解的最小秩、最大秩,及最小秩解的一般表达式.另外,推出了中心对称最小秩解集合中与给定矩阵的最佳逼近解.

By applying a new metod, the necessary and sufficient conditions for the existence of central-symmetric solutions and the general expression to the matrix equation AXB=C are obtained;according to the quotient singular value decomposition of matrix pairs and generalized inverses, minimal or maximal rank of the solutions and the general expression of the minimal rank solutions are givened. In addition,for the minimal rank solution set,the expression of the optimal approximation solution to a given matrix is deduced.