An Inverse Problem of Matrix with Fixed Row and Column Sums and its Application

In this paper, we study the inverse problem of an n×n matrix with fixed row and column sums by using the singular value decomposition (SVD) of a matrix. The least-square solutions of an n×n matrix with fixed row and column sums are studied. The necessary and sufficient conditions for the existence of the symmetric solutions and the expressions for the inverse problem of a matrix AX=B are established. In addition, the problem of using matrix with fixed row and column sums to construct the optimal approximation to a given matrix is discussed, and the expression of the solution is provided. The algorithms are proposed and applications to the theory of electric net are illustrated by examples.