A geometrical approach to the maximal corank problem in the analysis of linear relations

In this paper, a novel approach is provided to an important but unsolved mathematical problem that occurs in a wide variety of applications: Given a symmetric positive definite n??n matrix ??, determine all diagonal nonnegative matrices ??~ so that the difference matrix ??= ??- ??~ is nonnegative definite and its rank is minimal. In this paper, we explore the geometrical properties of the solution vectors x satisfying ??.x=0. New concepts such as orthant and null invariance are introduced. The results in this paper are of key importance in the analysis of noisy linear equations and factor analysis. They Hill lead to a complete geometrical characterization of the solution set, which will be described in a forthcoming paper.