Basic Soules matrices and their applications Original Research Article

In this paper we study the properties of the basic Soules matrices in image which are a special subclass of the n × n Soules matrices generated via the basic Soules basis. The basic Soules basis has the sign pattern image and it corresponds to the vector image of all 1’s. The basic Soules matrices are up to a multiple by a positive scalar, symmetric and doubly stochastic. We begin by investigating the permanents of basic Soules matrices. Next, for a nonsingular basic Soules matrix image, we show that the matrix A ring operator A−1, which is known to be a nonsingular M-matrix, has a basic Soules basis of eigenvectors. Furthermore, we obtain explicit formulas for the eigenvalues of A ring operator A−1 in terms of the eigenvalues of A. Finally, let image be a basic Soules matrix of spectral radius 1 and set Q = I − A. By investigating the sign pattern of the off-diagonal entries of the group inverse Q# of Q, we determine when the Perron root is a concave function in each of the off-diagonal entries at A.