The Congruent Centralizer of the Jordan Block

Abstract

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 \( {C}_J^{\ast } \) 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 \( {C}_J^{\ast } \) for an arbitrary n and then gives complete descriptions of the groups \( {C}_J^{\ast } \) for n = 2, 3, 4, 5.