Basic Soules matrices and their applications

In this paper we study the properties of the basic Soules matrices in Rn,n 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 N and it corresponds to the vector en∈Rn 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 A∈Rn,n, we show that the matrix A ∘ 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 ∘ A−1 in terms of the eigenvalues of A. Finally, let A∈Rn,n 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.