The generalized orthogonal solutions of the matrix inverse problem AX=B and optimal approximation

In this paper, the generalized orthogonal solutions to the matrix inverse problem $AX=B$ and associated optimal approximation problem are considered. The properties and structure of generalized orthogonal matrices are given, the relationships between the generalized orthogonal matrices and the orthogonal matrices are discussed. Necessary and sufficient conditions that the matrix inverse problem $AX=B$ is solvable, the general expression of solution, and it's procrustes problem are discussed. Moreover, the corresponding optimal approximation solutions are given. Finally, we give the algorithms and corresponding computational examples.