The change of similarity invariants under row perturbations

The problem of the possible similarity classes of all the matrices obtained by small perturbations on some rows of a given complex square matrix is under consideration. Some necessary conditions on these similarity classes are provided. The near converse problem is also studied: under what conditions can a given collection of polynomials or partitions be the invariant factors or the Weyr characteristic of a matrix obtained by perturbing some rows of a given matrix? A complete answer is provided in some particular cases. The obtained conditions can all be derived from the previously mentioned necessary conditions.