  We present a novel finite-matrix formulation of gauge theories on a
non-commutative torus. Unlike the previous formulation based on a map from a
square matrix to a field on a discretized torus with periodic boundary
conditions, our formulation is based on the algebraic characterization of the
configuration space. This enables us to describe the twisted boundary
conditions in terms of finite matrices and hence to realize the Morita
equivalence at a fully regularized level. Matter fields in the fundamental
representation turn out to be represented by rectangular matrices for twisted
boundary conditions analogously to the matrix spherical harmonics on the fuzzy
sphere with the monopole background. The corresponding Ginsparg-Wilson Dirac
operator defines an index, which can be used to classify gauge field
configurations into topological sectors. We also perform Monte Carlo
calculations for the index as a consistency check. Our formulation is expected
to be useful for applications of non-commutative geometry to various problems
related to topological aspects of field theories and string theories.
