**In minisymposium: Matrix Equations**

If $T$ is the transition matrix of a finite irreducible Markov chain and ${b}$ is a given vector in the image of $I-T$, the Poisson equation $(I-T){x}={b}$ has a unique solution, up to an additive constant, given by $ {x}=(I-T)^{\#}{b} +\alpha {1}$, for any $\alpha$ scalar, where ${1}$ is the vector of all ones and $H^{\#}$ denotes the group inverse of $H$. This does not hold when the state space is infinite.

We consider the Poisson equation $(I-P){u}={g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, ${g}$ is a given infinite–dimensional vector and ${u}$ is the unknown vector.

By using the block tridiagonal and block Toeplitz structure of $P$, we recast the problem in terms of a set of matrix difference equations. We provide an explicit expression of the general solution, relying on the properties of Jordan triples of matrix polynomials and on the solutions of suitable quadratic matrix equations. The uniqueness and boundedness of the solutions are discussed, according to the properties of the right–hand side ${g}$.