Detecting stable matrices
The author establishes the relation between stable matrices, as being interesting to economists, and their entries. A matrix is called stable if the real parts of its characteristic roots are negative. More generally, a matrix is Hicksian stable if its odd order principal minors are negative and its even order principal minors are positive. The topic of this paper is the means of assessing if a matrix satisfies either or both of these stability conditions. The major contribution to this issue is the concept of a matrix with a quasi-dominant diagonal and its conditions on the relationships between the (absolute) sizes of the main diagonal elements and the sums of its off-diagonal elements. Furthermore restrictive conditions were developed for the qualitative or sign stability for the case that only the signs of the elements are known. The approach taken in this paper is to extend the conditions on the signs of cycles in the corresponding signed directed graph to conditions on the sum of the values of cycles. Specifications of standard forms for the matrix to have a quasi-dominant diagonal and the related concepts are briefly provided. Conditions on the values of the cycles are related to concepts of stability of the Jacobian matrix of a linear approximation of the system of differential equations of an input-output model. Some extensions of the results are also given.
scientific article; zbMATH DE number 1123704