Matrix Product Eigenstates for One-Dimensional Stochastic Models and Quantum Spin Chains

We study a Hamiltonian with a two site interaction which describes either a one-dimensional stochastic $m$-state model or a quantum spin $(m-1)/2$ chain. We show that all its eigenstates can be represented as matrix product states, i.e. by means of operators and vectors in an auxiliary space which fulfill an algebra defined by the Hamiltonian.