  Hyperbolic formulations of the equations of motion are essential technique
for proving the well-posedness of the Cauchy problem of a system, and are also
helpful for implementing stable long time evolution in numerical applications.
We, here, present three kinds of hyperbolic systems in the Ashtekar formulation
of general relativity for Lorentzian vacuum spacetime. We exhibit several (I)
weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric
hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's
original equations form a weakly hyperbolic system. We discuss how gauge
conditions and reality conditions are constrained during each step toward
constructing a symmetric hyperbolic system.
