The rank and eigenvalues of main diagonal perturbed matrices

An (n n)-matrix T is in class Mk if and only if T can be written as the sum of a diagonal matrix and a matrix of rank k. It is shown that The paper concentrates on the class M 1 and the eigenvalues of its members. For example, it is shown that in case all the main diagonal elements have the same sign then all the eigenvalues, except one, are in the interval determined by the maximum and minimum main diagonal element of the corresponding diagonal matrix. Applications for the input-output Leontief model and oligopoly-like games are given. The paper concludes with four research problems.