Symmetric hyperbolic system in the ashtekar formulation

We present a first-order symmetric hyperbolic system in the Ashtekar formulation of general relativity for vacuum spacetime. We add terms from the constraint equations to the evolution equations with appropriate combinations, which is the same technique used by Iriondo, Leguizamón, and Reula [Phys. Rev. Lett. 79, 4732 (1997)]. However, our system is different from theirs in that we primarily use Hermiticity of a characteristic matrix of the system to characterize our system symmetric, discuss the consistency of this system with a reality condition, and show the characteristic speeds of the system.