A note on the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation (Q1855425)

Consider the nonsymmetric algebraic Riccati equation (ARE) \(XCX-XD-AX+B=0\) where \(A\in \mathbb{R}^{m\times m},\;B\in \mathbb{R}^{m\times n},C\in \mathbb{R}^{n\times m},\) and \(D\in \mathbb{R}^{n\times n}\) and \[ K=\begin{pmatrix} D & -C \\ -B & A \end{pmatrix} \] is an \(M\)-matrix. Then the paper proves that if the matrix \(K\) is an irreducible \(M\)-matrix, then the minimal nonnegative solution of the ARE, let \(S\), is positive i.e. \(S>0\). This assumption (\(S>0\)) is needed for the nonsymmetric ARE arising in the Wiener-Hopf factorization of Markov chains.