Experiences in Symbolic Computations for Matrix Problems

We review our recent results concerning several computer algebra aspects of determining canonical forms, performing a decomposition and deciding the isomorphism question for matrix problems. We consider them in the language of finite dimensional modules over algebra and the language of square block matrices with an action of elements from some sub algebra of the full matrix algebra. We present an efficient (polynomial-time) improvement of classical Bongartz's algorithm for determining a maximal common direct summand of modules, and its application to solving the isomorphism problem. The improved algorithm recently became a part of QPA package ver. 1.07 for GAP. We also study the behaviour of Belitskii's algorithm for determining certain canonical form on a class of square block matrices, especially for matrix problems associated with a poset. Both problems can be considered as a highly generalized classical Jordan problem for square matrices.