Research On The Least Squares Problems For A Quaternion Tensor Equation

In this paper,we mainly investigate the least squares problems of quaternion tensor equa-tion A*_NX=B,where the symbol*_Nis the Einstein product of two tensors.Firstly,we transform the quaternion tensor equation A*_NX=B into complex(real)tensor equations by using the complex representation of quaternion tensors,obtain the expression of least squares solutions by using the Moore-Penrose generalized inverse of tensors,and give the correspond-ing numerical algorithm and numerical examples.Secondly,we transform the quaternion tensor equation A*_NX=B into a real tensor equation by combining a product for real tensors and vectors,obtain the Hermitian least square solutions and the supersymmetric least square solu-tions by using the Moore-Penrose generalized inverse of matrices,and give the corresponding numerical algorithm and numerical examples.In the first chapter,we introduce the research background,purpose and mathematical sym-bols of this paper,and basic concepts and properties of quaternion tensors.In the second chapter,we study the least square solution with the least norm,the least square pure imaginary quaternion solution with the least norm,and the least square real solu-tion with the least norm for the quaternion tensor equation A*_NX=B by using the complex representation of quaternion tensors,and give the numerical algorithms and numerical exam-ples.In the third chapter,we introduce a product of real tensor and vector,study the least square Hermitian solution for the quaternion tensor equation A*_NX=B,and give the numerical algorithm and numerical examples.In the forth chapter,we study the least square supersymmetric solution for quaternion tensor equation A*_NX=B by using the product of real tensors and vectors,and give the numerical algorithm and numerical examples.In the fifth chapter,we summarize the research results and look forward to the direction of future efforts.