四元数Sylvester张量方程A*_(N)x-y*_(N)B=C的混合结构解

Mixed Structural Solutions of Quaternion Sylvester Tensor Equation A*_(N)x-y*_(N)B=C

讨论了Einstein积意义下四元数Sylvester张量方程A*_(N)x-y*_(N)B=C的一种混合结构解,其中x,y分别是未知Hermite张量与Skew-Hermite张量.利用四元数张量共轭转置的性质,将原结构方程转化为一个等价的无约束张量方程组,再用四元数张量的Moore-Penrose广义逆,获得等价方程组可解的充要条件及其通解表达式,从而得到原方程的混合结构解.特别地,导出了张量方程A*_(N)x=C与y*_(N)B=-C分别具有Hermite张量解与Skew-Hermite张量解的条件及其解表达式.数值算例检验了所得结果的正确及可行性.

The mixed structural solutions of the quaternion Sylvester tensor equation A*_(N)x-y*_(N)B=C in the sense of Einstein product is discussed,where x is unknown Hermite tensor,and y is unknown skew-Hermite tensor respectively.By using the property of the conjugate transpose of quaternion tensor,the original structural equation can be transformed into an equivalent unconstrained tensor equation group.Then,by using the Moore-Penrose generalized inverse of quaternion tensor,we establish some necessary and sufficient solvability conditions and general solution expression of equivalent unconstrained tensor equation group,thus,some necessary and suficient solvability conditions and concrete expressions of the original equation with the proposed mixed structure solutions can be obtained.In particular,the conditions and expressions of the tensor equations A*_(N)x=C and y*_(N)B=-C with Hermite and skew-Hermite tensor solutions are derived.A numerical example is given to verify the correctness and feasibility of the results.