  We consider matrices with entries in a local ring, Mat(m,n,R). Fix a group
action, G on Mat(m,n,R), and a subset of allowed deformations, \Sigma\subseteq
Mat(m,n,R). The standard question of Singularity Theory is the
finite-(\Sigma,G)-determinacy of matrices. Finite determinacy implies
algebraizability and is equivalent to a stronger notion: stable
algebraizability.
  In our previous work this determinacy question was reduced to the study of
the tangent spaces to \Sigma and to the orbit, T_{(\Sigma,A)}, T_{(GA,A)} , and
their quotient, the tangent module to the miniversal deformation. In
particular, the order of determinacy is controlled by the annihilator of this
tangent module.
  In this work we study this tangent module for the group action GL(m,R)\times
GL(n,R) on Mat(m,n,R) and various natural subgroups of it. We obtain
ready-to-use criteria of determinacy for deformations of (embedded) modules,
(skew-)symmetric forms, filtered modules, filtered morphisms of filtered
modules, chains of modules etc.
