On the Kronecker Problem and related problems of Linear Algebra

We consider some classification problems of Linear Algebra related closely to the classical Kronecker Problem on pairs of linear maps between two finite-dimensional vector spaces. As shown by Djoković and Sergeichuk, the Kronecker’s solution is extended to the cases of pairs of semilinear maps and (more generally) pseudolinear bundles respectively. Our objective is to deal with the semilinear case of the Kronecker Problem, especially with its applications. It is given a new short solution both to this case and to its contragredient variant. The biquadratic matrix problem is investigated and reduced in the homogeneous case (in characteristic ≠2) to the semilinear Kronecker Problem. The integer matrix sequence Θn and Θ-transformation of polynomials are introduced and studied to get a simplified canonical form of indecomposables for the mentioned homogeneous problem. Some applications to the representation theory of posets with additional structures are presented.