  We consider the actions of different groups G on the space M of m x n
matrices with entries in the formal power series ring K[[x1,..., xs]], K an
arbitrary field. G acts on M by analytic change of coordinates, combined with
the multiplication by invertible matrices from the left, the right or from both
sides, respectively. This includes right and contact equivalence of functions
and mappings, resp. ideals. A is called finitely G-determined if any matrix B,
with entries of A-B in <x1,...,xs>^k for some k, is contained in the G-orbit of
A. The purpose of this paper is to present algorithms for checking finite
determinacy, to compute determinacy bounds and to compute the image of the
tangent map to the orbit map G -> GA, which we call the tangent image to the
orbit GA. The tangent image is contained in the tangent space to the orbit GA
and we apply the algorithms to prove that both spaces may be different if the
field K has positive characteristic, even for contact equivalence of functions.
This fact had been overlooked by several authors before. Besides this
application, the algorithms of this paper may be of interest for the
classification of singularities in arbitrary characteristic.
