Abstract: Using the singular value decomposition (SVD) of a matrix and the canonical correction decomposition (CCD) of matrices, this paper establishes the necessary and sufficient conditions for the existence of the doubly center solutions and presents the expressions for the inverse problem of a matrix under a submatrix constraint. Moreover, the optimal approximation problem to a given matrix in the solution set is considered, and the optimal approximation solution is obtained.

Key words: submatrix constraint, doubly center matrices, inverse problem, optimal approximation, canonical correction decomposition