  A block diagonal preconditioner with the minimal residual method and a block
triangular preconditioner with the generalized minimal residual method are
developed for Hu-Zhang mixed finite element methods of linear elasticity. They
are based on a new stability result of the saddle point system in
mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the
mass matrix which is easy to invert while the displacement it is spectral
equivalent to Schur complement. A fast auxiliary space preconditioner based on
the $H^1$ conforming linear element of the linear elasticity problem is then
designed for solving the Schur complement. For both diagonal and triangular
preconditioners, it is proved that the conditioning numbers of the
preconditioned systems are bounded above by a constant independent of both the
crucial Lam\'e constant and the mesh-size. Numerical examples are presented to
support theoretical results. As byproducts, a new stabilized low order mixed
finite element method is proposed and analyzed and superconvergence results of
Hu-Zhang element are obtained.
