Miniversal deformations of chains of linear mappings

Abstract

V.I.Arnold [Russian Math. Surveys, 26 (no. 2), 1971, pp. 29-43] gave a miniversal deformation of matrices of linear operators; that is, a simple canonical form, to which not only a given square matrix \(A\), but also the family of all matrices close to \(A\), can be reduced by similarity transformations smoothly depending on the entries of matrices. We study miniversal deformations of quiver representations and obtain a miniversal deformation of matrices of chains of linear mappings \[ V_1 \,\frac{}{\qquad}\, V_2\,\frac{}{\qquad}\, \cdots \,\frac{}{\qquad}\, V_t\,, \] where all \(V_i\) are complex or real vector spaces and each line denotes \(\longrightarrow\) or \(\longleftarrow\).