On a quadratic matrix equation associated with an M‐matrix

We study the quadratic matrix equation X² − EX − F = 0, where E is diagonal and F is an M‐matrix. Quadratic matrix equations of this type arise in noisy Wiener–Hopf problems for Markov chains. The solution of practical interest is a particular M‐matrix solution. The existence and uniqueness of M‐matrix solutions and numerical methods for finding the desired M‐matrix solution are discussed by transforming the equation into an equation that belongs to a special class of non‐symmetric algebraic Riccati equations (AREs). We also discuss the general non‐symmetric ARE and describe how we can use the Schur method to find the stabilizing or almost stabilizing solution if it exists. The desired M‐matrix solution of the quadratic matrix equation (a special non‐symmetric ARE by itself) turns out to be the unique stabilizing or almost stabilizing solution.