The Structure of Qualitatively Determinate Relationships
给出了仅根据系数矩阵和右端项的符号判断线性方程组解变量符号的充要条件，对经济学中基于符号而非具体函数形式的实证分析有重要意义。
This paper presents the necessary and sufficient conditions for determining the signs of the solution variables of a system of linear equations based only upon a knowledge of the signs of the coefficient matrix and the signs of the right hand side variables. This problem was initially formulated in economics due to the idea that the signs of an equation's derivatives might have a stronger empirical basis than that of a particular functional form. A new interest in qualitative problems has arisen in connection with the need to develop analytic measures in order to better manage the understanding and use of large, computer-based mathematical systems. The conditions for the qualitative determinancy of nonhomogeneous systems are developed in terms of a small number of necessary conditions which are jointly sufficient. Algorithmic approaches are given for testing a given system for qualitative determinancy. For nonhomogeneous systems algorithms are given for constructing all possible qualitatively determinate systems of a given size. For the homogeneous case conditions are also given for the qualitative invertibility of the (irreducible) coefficient matrix. These conditions are then related to the problem of partially qualitatively determinate systems and the signs in the qualitative inverse of a matrix.