On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.