On the class of matrices with rows that weakly decrease cyclicly from the diagonal

Abstract

We consider n×n real-valued matrices A=(aij) satisfying aii≥ai,i+1≥…≥ain≥ai1≥…≥ai,i−1 for i=1,…,n. With such a matrix A we associate a directed graph G(A). We prove that the solutions to the system A⊤x=λe, with λ∈R and e the vector of all ones, are linear combinations of ‘fundamental’ solutions to A⊤x=e and vectors in ker⁡A⊤, each of which is associated with a closed strongly connected component (SCC) of G(A). This allows us to characterize the sign of det⁡A in terms of the number of closed SCCs and the solutions to A⊤x=e. In addition, we provide conditions for A to be a P-matrix.