  We study a systematic derivation of four dimensional $\mathcal{N}=1$
supersymmetric effective theory from ten dimensional non-Abelian
Dirac-Born-Infeld action compactified on a six dimensional torus with magnetic
fluxes on the D-branes. We find a new type of matter K\"{a}hler metric while
gauge kinetic function and superpotential are consistent with previous studies.
For the ten dimensional action, we use a symmetrized trace prescription and
focus on the bosonic part up to $\mathcal{O}(F^4)$. In the presence of the
supersymmetry, four dimensional chiral fermions can be obtained via index
theorem. The new matter K\"{a}hler metric is independent of flavor but depends
on the fluxes, 4D dilaton, K\"{a}hler moduli and complex structure moduli, and
will be always positive definite if an induced Ramond-Ramond charge of the
D-branes on which matters are living are positive. We read the superpotential
from an F-term scalar quartic interaction derived from the ten dimensional
action and the contribution of the new matter K\"{a}hler metric to the scalar
potential which we derive turns out to be consistent with the supergravity
formulation.
