The congruent centralizer of the Jordan block

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 CA 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 CJ∗ even in the simplest case of the Jordan block J = Jn(0) with zeros on the principal diagonal. This paper presents certain facts concerning the structure of matrices in CJ∗ for an arbitrary n and then gives complete descriptions of the groups CJ∗ for n =2, 3, 4, 5. © 2017 Springer Science+Business Media New York.