Some Problems On The Product-type Generalized Inverse Of Matrices And Tensors

The generalized inverse of matrices and tensors can be applied to the solution process of matrix equations and tensor equations.In the actual process of solving the matrix equations or tensor equations,due to the possibility of errors in the specific operation process,the real solution and the calculated solution are different,so it is meaningful to consider the perturbation of the generalized inverse.In recent research,people have promoted some concepts and properties of generalized inverses from matrix space to 2N-order tensor space.The product-type generalized inverse in this paper includes DMP inverse and core-EP inverse.In this paper,we first analyze the DMP inverse of operator matrices under two-sided and one-sided disturbance conditions.Then,the DMP inverse is defined via the Einstein product,and the block form of the tensor via the Einstein product is used to obtain the decomposition of DMP inverse.And the perturbation analysis of the DMP inverse of tensors via the Einstein product is analyzed.Finally,the block form of the tensor via the Einstein product is used to obtain the decomposition of core-EP inverse.And the perturbation analysis of the core-EP inverse of tensors via the Einstein product is analyzed.