  Most of the exact solutions of quantum one-dimensional Hamiltonians are
obtained thanks to the success of the Bethe ansatz on its several formulations.
According to this ansatz the amplitudes of the eigenfunctions of the
Hamiltonian are given by a sum of permutations of appropriate plane waves. In
this paper, alternatively, we present a matrix product ansatz that asserts that
those amplitudes are given in terms of a matrix product. The eigenvalue
equation for the Hamiltonian define the algebraic properties of the matrices
defining the amplitudes. The existence of a consistent algebra imply the exact
integrability of the model. The matrix product ansatz we propose allow an
unified and simple formulation of several exact integrable Hamiltonians. In
order to introduce and illustrate this ansatz we present the exact solutions of
several quantum chains with one and two global conservation laws and periodic
boundaries such as the XXZ chain, spin-1 Fateev-Zamolodchikov model,
Izergin-Korepin model, Sutherland model, t-J model, Hubbard model, etc.
Formulation of the matrix product ansatz for quantum chains with open ends is
also possible. As an illustration we present the exact solution of an extended
XXZ chain with $z$-magnetic fields at the surface and arbitrary hard-core
exclusion among the spins.
