  Solving the 4-d Einstein equations as evolution in time requires solving
equations of two types: the four elliptic initial data (constraint) equations,
followed by the six second order evolution equations. Analytically the
constraint equations remain solved under the action of the evolution, and one
approach is to simply monitor them ({\it unconstrained} evolution). Since
computational solution of differential equations introduces almost inevitable
errors, it is clearly "more correct" to introduce a scheme which actively
maintains the constraints by solution ({\it constrained} evolution). This has
shown promise in computational settings, but the analysis of the resulting
mixed elliptic hyperbolic method has not been completely carried out. We
present such an analysis for one method of constrained evolution, applied to a
simple vacuum system, linearized gravitational waves.
  We begin with a study of the hyperbolicity of the unconstrained Einstein
equations. (Because the study of hyperbolicity deals only with the highest
derivative order in the equations, linearization loses no essential details.)
We then give explicit analytical construction of the effect of initial data
setting and constrained evolution for linearized gravitational waves. While
this is clearly a toy model with regard to constrained evolution, certain
interesting features are found which have relevance to the full nonlinear
Einstein equations.
