一类张量方程的可解性及其最佳逼近问题

Solvability Conditions for a Class of Tensor Equations and Associated Optimal Approximation Problems

研究了张量方程A*_(n)X=B具有Hermitian解X的可解性问题,其中*_(n)表示张量的Einstein积.利用张量Moore-Penrose广义逆的性质,得到了该方程具有Hermitian解的充要条件及其通解表达式.同时,在张量的Frobenius范数意义下,考虑了对于任意给定张量的最佳逼近问题,得到了它的唯一解表达式.最后,通过数值例子说明了结论的可行性.

This paper is concerned with the solution to the tensor equation A*_(n)X=B with Hermitian X,where represents the Einstein product.Depending on the properties of Moore-Penrose generalized inverses of tensors,the solvability conditions for the existence of the Hermitian solution to the above tensor equation as well as its general solution have been derived.Meanwhile,the associated tensor approximation problem for any given tensor has been considered and the unique solution has been given.Finally,the performed numerical results demonstrate the feasibility of the proposed results.