  A new classical theory of gravitation within the framework of general
relativity is presented. It is based on a matrix formulation of
four-dimensional Riemann-spaces and uses no artificial fields or adjustable
parameters. The geometrical stress-energy tensor is derived from a matrix-trace
Lagrangian, which is not equivalent to the curvature scalar R. To enable a
direct comparison with the Einstein-theory a tetrad formalism is utilized,
which shows similarities to teleparallel gravitation theories, but uses complex
tetrads. Matrix theory might solve a 27-year-old, fundamental problem of those
theories (sec. 4.1). For the standard test cases (PPN scheme,
Schwarzschild-solution) no differences to the Einstein-theory are found.
However, the matrix theory exhibits novel, interesting vacuum solutions.
