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We consider matrices with entries in a local ring, Mat(m,n,R). Fix a groupaction, G on Mat(m,n,R), and a subset of allowed deformations, \Sigma\subseteqMat(m,n,R). The standard question of Singularity Theory is thefinite-(\Sigma,G)-determinacy of matrices. Finite determinacy impliesalgebraizability and is equivalent to a stronger notion: stablealgebraizability.  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.  In this work we study this tangent module for the group action GL(m,R)\timesGL(n,R) on Mat(m,n,R) and various natural subgroups of it. We obtainready-to-use criteria of determinacy for deformations of (embedded) modules,(skew-)symmetric forms, filtered modules, filtered morphisms of filteredmodules, chains of modules etc.

Finite determinacy of matrices over local rings. Tangent modules to the miniversal deformation for R-linear group actions