  This is a work in two parts in which we show how to solve a large class of
Lindblad master equations for non-interacting particles on $L$ sites. In part I
we concentrate on bosonic particles. We show how to reduce the problem to
diagonalizing an $L \times L$ non-Hermitian matrix. In particular, for boundary
dissipative driving of a uniform chain, the matrix is a tridiagonal bordered
Toeplitz matrix which can be solved analytically for the normal master modes
and their relaxation rates (rapidities). In the regimes in which an analytical
solution cannot be found, our approach can still provide a speed-up in the
numerical evaluation. We use this numerical method to study the relaxation gap
at non-equilibrium phase transitions in a boundary driven bosonic ladder with
synthetic gauge fields. We conclude by showing how to construct the
non-equilibrium steady state.
