In the dissertation we study finite-dimensional multiparameter eigenvalue problems. The main objects considered are multiparameter systems, i.e., systems of n linear n-parameter pencils. To a multiparameter system we associate an n-tuple of commuting matrices. The main problem considered is to describe a basis for the root subspaces of an associated system in terms of the underlying multiparameter system.

In the present thesis, we study the possible Jordan canonical forms for a pair of commuting nilpotent matrices: (1) We give the exact upper bound for the index of nilpotency of a nilpotent matrix commuting with a given nilpotent matrix $B$. (2) We characterize all possible pairs of Jordan canonical forms for nilpotent matrices $A$ and $B$ such that $AB = BA = 0$. We characterize all matrices that can be written as a product of two commuting nilpotent matrices.(3) We give complete lists of Jordan canonical forms of nilpotent matrices, commuting with (a) a nilpotent matrix with only one Jordan block, (b) a square-zero matrix, (c) an $n \times n$ nilpotent matrix, $n \le 8$. (5) We give a large set of possible Jordan canonical forms of a nilpotent matrix, commuting with a nilpotent matrix with two Jordan blocks.