On a quadratic matrix equation associated with an M‐matrix

We study the quadratic matrix equation X 2 - 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.

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