张量方程A*_(N)X=B的Hermitian解与半正定解

Hermitian Solution and Positive Semidefinite Solution of Tensor Equation A*_(N)X=B

在张量的Einstein积下分别研究了张量方程A*_(N)X=B的Hermitian解与半正定解。根据方程解的结构,分别求得方程的特解与对应齐次方程的通解。并给出了这两种解存在的充要条件及通解的显式表达式。张量方程的Hermitian解与半正定解存在的充要条件表明,矩阵方程的Hermitian解与半正定解的性质与通解的显式表达式可以推广到张量上。

Hermitian solution and positive semidefinite solution to tensor equation can be regarded as an extension of the matrix equation.According to the structure of the solution,a special solution to the tensor equation is found,and the general solution to the homogeneous equation is established corresponding to the tensor equation.The necessary and sufficient conditions for the existence and the explicit expressions for these two solutions to the tensor equation are presented.The necessary and sufficient conditions for the existence of Hermitian solution and positive semi-definite solution of tensor equations show that the properties of Hermitian and positive semi-definite solutions of matrix equation and the explicit expressions of general solutions can be extended to tensor equations.