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We consider matrices with entries in a local ring, Mat(m,n;R). Fix an actionof group G on Mat(m,n;R), and a subset of allowed deformations, \Sigma inMat(m,n;R). The standard question (along the lines of Singularity Theory) isthe finite-(\Sigma,G)-determinacy of matrices.  In our previous work this determinacy question was reduced to the study ofthe tangent spaces to \Sigma and to the orbit, T_{(\Sigma,A)}, T_{(GA,A)}, andtheir quotient: the tangent module to the miniversal deformation. Inparticular, the order of determinacy is controlled by the annihilator of thistangent module.  Then we have studied this tangent module for the group action GL(m,R)\timesGL(n,R) on Mat(m,n;R) and for various natural subgroups of it. These areR-linear group actions.  In the current work we study this tangent module for group actions thatinvolve the automorphisms of the ring, or, geometrically, group-actions thatinvolve the local coordinate changes. (These actions are not R-linear.) Weobtain various bounds on the support of this module. This gives ready-to-usecriteria of determinacy for matrices, (embedded) modules and (skew-)symmetricforms.

Finite determinacy of matrices over local rings.II. Tangent modules to  the miniversal deformations for group-actions involving the ring  automorphisms