  We propose a nonperturbative framework for the O(32) type I open and closed
string theory. The short distance degrees of freedom are bosonic and fermionic
hermitian matrices belonging respectively to the adjoint and fundamental
representations of the special unitary group SU(N). We identify a closed matrix
algebra at finite N which corresponds to the Lorentz, gauge, and supersymmetry
algebras of the large N continuum limit. The planar reduction of our matrix
theory coincides with the low energy spacetime effective action of the d=10
type I O(32) unoriented open and closed string theory. We show that matrix
T-duality transformations can yield a nonperturbative framework for the T-dual
type I' closed string theory with 32 D8branes. We show further that under a
strong-weak coupling duality transformation the large N reduced action
coincides with the low energy spacetime effective action of the d=10 heterotic
string, an equivalence at leading order in the inverse string tension and with
either gauge group Spin(32)/Z2 or E8xE8. Our matrix formalism has the potential
of providing a nonperturbative framework encapsulating all of the weak coupling
limits of M theory.
