In this work, we study the classification of pairs of square matrices $(A,B)$ up to simultaneous similarity, with a particular focus on the class of commuting nilpotent matrix pairs. We introduce the Jordan and Weyr canonical forms, as well as the set of matrices known as the reduced matrix algebra, which consists of all matrices commuting with the Weyr matrix. We examine Belitskii's algorithm, which constructs canonical representatives of simultaneous similarity classes with respect to a given reduced matrix algebra, and further describe the process of determining whether two matrix pairs are simultaneously similar. Special attention is devoted to pairs of commuting nilpotent matrices. For pairs in which the first matrix consists of two Jordan blocks of different sizes, we can explicitly determine the canonical representatives. The theoretical results are supported with detailed computations and examples. We conclude by describing the full set of representatives for $4\times4$ commuting nilpotent pairs.
|
