Matrix geometries and fuzzy spaces as finite spectral triples

A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for the Dirac operator. Examples include fuzzy spaces defined as real spectral triples. Fuzzy 2-spheres are investigated in detail, and it is shown that the fuzzy analogues correspond to two spinor fields on the commutative sphere. In some cases it is necessary to add a mass mixing matrix to the commutative Dirac operator to get a precise agreement for the eigenvalues.