  Instanton matrix models (IMM) for two dimensional string theories are
obtained from the matrix quantum mechanics (MQM) of the T-dual theory. In this
paper we study the connection between the IMM and MQM, which amounts to
understand T-duality from the viewpoint of matrix models. We show that type 0A
and type 0B matrix models perturbed by purely closed string momentum modes (or
purely winding modes) have the integrable structure of Toda hierarchies,
extending the well known results for c=1 string. In particular, we show that
type 0A(0B) MQM perturbed by momentum modes has the same integrable structure
as type 0B(0A) MQM perturbed by winding modes, which is a nontrivial check of
the T-duality between the matrix models. The MQM deformed by NS-NS winding
modes are used to study type 0 string in 2D black holes. We also find an
intriguing connection between the IMM and the MQM via tachyon condensation. The
array of alternating D-instantons and anti-D-instantons separated at the
critical distance plays a key role in this picture. We discuss its implications
on sD-branes in two dimensional string theories.
