THE CONGRUENT CENTRALIZER OF THE JORDAN BLOCK

摘要

The congruent centralizer of a complex n × n matrix A is the set of n × n matrices Z such that Z~* AZ = A. This set is an analog of the classical centralizer in the case where the similarity relation on the space of n × n matrices is replaced by the congruence relation. The study of the classical centralizer C_a reduces to describing the set of solutions of the linear matrix equation AZ = ZA. The structure of this set is well known and is explained in many monographs on matrix theory. As to the congruent centralizer, its analysis amounts to a description of the solution set of a system of n2 quadratic equations for n2 unknowns. The complexity of this problem is the reason why there is still no description of the congruent centralizer C_J~* even in the simplest case of the Jordan block J = J_n(0) with zeros on the principal diagonal. This paper presents certain facts concerning the structure of matrices in C_J~* for an arbitrary n and then gives complete descriptions of the groups C_J~* for n = 2,3,4,5. Bibliography: 1 title.