  We review the construction of the Dirac operator and its properties in
Riemannian geometry and show how the asymptotic expansion of the trace of the
heat kernel determines the spectral invariants of the Dirac operator and its
index. We also point out that the Einstein-Hilbert functional can be obtained
as a linear combination of the first two spectral invariants of the Dirac
operator. Next, we report on our previous attempts to generalize the notion of
the Dirac operator to the case of Matrix Geometry, when instead of a Riemannian
metric there is a matrix valued self-adjoint symmetric two-tensor that plays a
role of a ``non-commutative'' metric. We construct invariant first-order and
second-order self-adjoint elliptic partial differential operators, that can be
called ``non-commutative'' Dirac operator and non-commutative Laplace operator.
We construct the corresponding heat kernel for the non-commutative Laplace type
operator and compute its first two spectral invariants. A linear combination of
these two spectral invariants gives a functional that can be considered as a
non-commutative generalization of the Einstein-Hilbert action.
