  Recent work has shown that unstable D-branes in two dimensional string theory
are represented by eigenvalues in a dual matrix model. We elaborate on this
proposal by showing how to systematically include higher order effects in
string perturbation theory. The full closed string state produced by a rolling
open string tachyon corresponds to a sum of string amplitudes with any number
of boundaries and closed string vertex operators. These contributions are
easily extracted from the matrix model. As in the AdS/CFT correspondence, the
sum of planar diagrams in the open string theory is directly related to the
classical theory in the bulk, i.e. sphere diagrams. We also comment on the
description of static D-branes in the matrix model, in terms of a solution
representing a deformed Fermi sea.
