  A manifestly Lorentz-covariant calculus based on two matrix-coordinates and
their associated derivatives is introduced. It allows formulating relativistic
field theories in any even-dimensional spacetime. The construction extends a
single-coordinate matrix formalism based on coupling spacetime coordinates with
the corresponding Gamma-matrices. A 2D matrix-calculus can be introduced for
each one of the structures, adjoint, complex and transposed acting on
Gamma-matrices. The adjoint structure works for spacetimes with (n,n) signature
only. The complex structure requires an even number of timelike directions. The
transposed structure is always defined. A further structure which can be
referred as "spacetime-splitting" is based on a fractal property of the
Gamma-matrices. It is present in spacetimes with dimension D=4n+2. The
conformal invariance in the matrix-approach is analyzed. A complex conjugation
is present for the complex structure, therefore in euclidean spaces, or
spacetimes with (2,2), (2,4) signature and so on. As a byproduct it is here
introduced an index which labels the classes of inequivalent Gamma-structures
under conjugation performed by real and orthogonal matrices. At least two
timelike directions are necessary to get more than one classes of equivalence.
Furthermore an algorithm is presented for iteratively computing D-dimensional
Gamma-matrices from the p and q dimensional ones where D=p+q+2. Possible
applications of the 2D-matrix calculus concern the investigation of
higher-dimensional field theories with techniques borrowed from 2D-physics.
