A　block-diagonal　preconditioner　with　the　minimal　residual　method　and　an　approximate　block-factorization　preconditioner　with　the　generalized　minimal　residual　method　are　developed　for　Hu-Zhang　mixed　finite　element　methods　for　linear　elasticity.　They　are　based　on　a　new　stability　result　for　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　for　the　displacement　it　is　spectral　equivalent　to　the　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　Lame　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　for　the　Hu-Zhang　element　are　obtained.