Introducing the Class of SemiDoubly Stochastic Matrices: A Novel Scaling Approach for Rectangular Matrices

It is easy to verify that if A is a doubly stochastic matrix, then both its normal equations A A T and A T A are also doubly stochastic, but the reciprocal is not true. In this paper, we introduce and analyze the complete class of nonnegative matrices whose normal equations are doubly stochastic. This class contains and extends the class of doubly stochastic matrices to the rectangular case. In particular, we characterize these matrices in terms of their row and column sums and provide results regarding their nonzero structure. We then consider the diagonal equivalence of any rectangular nonnegative matrix to a matrix of this new class, and we identify the properties for such a diagonal equivalence to exist. To this end, we present a scaling algorithm and establish the conditions for its convergence. We also provide numerical experiments to highlight the behavior of the algorithm in the general case.