线性矩阵方程的斜Hermit{P,k+1}Hamilton解

The skew-Hermitian{P,k+1} Hamiltonian solutions of a linear matrix equation

给定矩阵P∈C^(n×n)且P^(*)=-P=P^(k+1).考虑了矩阵方程AX=B存在斜Hermite{P,k+1}(斜)Hamilton解的充要条件,并给出了解的表达式.进一步,对于任意给定的矩阵∈C^(n×n),给出了使得Frobenius范数‖Ã-A‖取得最小值的最佳逼近解Ã∈C^(n×n).当矩阵方程AX=B不相容时,给出了斜Hermite{P,k+1}(斜)Hamilton最小二乘解,在此条件下,给出了对于任意给定矩阵的最佳逼近解.最后给出一些数值实例.

GivenP∈C^(n×n) and P=-P=P^(k+1),we consider the necessary and sufficient conditions such that the matrix equation AX=B is consistent with the skew-Hermitian{P,k+1}(skew-)Hamiltonian structural constraint.Then,the corresponding expressions of the constraint solutions are also obtained.For any given matrixÃ∈C^(n×n),we present the optimal approximate solutionÃ∈C^(n×n) such that‖-‖is minimized in the Frobenius norm sense.If the matrix equation AX=B is not consistent,its least-squares skew-Hermitian{P,k+1}(skew-)Hamiltonian solutions are given.Under the least-square sense,we consider the best approximate solutions to any given matrix.Finally,some illustrative experiments are also presented.