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We show that a pair of almost commuting self-adjoint, symmetric matrices isclose to a pair of commuting self-adjoint, symmetric matrices (in a uniformway). Moreover we prove that the same holds with self-dual in place ofsymmetric. The notion of self-dual Hermitian matrices is important in physicswhen studying fermionic systems that have time reversal symmetry. Since asymmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin'sfamous theorem on almost commuting matrices. Similarly the self-dual case givesa version for matrices over the quaternions.  We prove analogous results for element of real C^*-algebras of "low rank." Inparticular, these stronger results apply to paths of almost commuting Hermitianmatrices that are real or self-dual. Along the way we develop a theory ofsemiprojectivity for real C^*-algebras.

Almost commuting self-adjoint matrices --- the real and self-dual cases