On the congruent centralizer of the Jordan block

Kh. D. Ikramov

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIX, Tome 453 (2016), pp. 85-95

Résumé

The congruent centralizer of a complex $n\times n$ matrix $A$ is the set of $n\times n$ 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 $n\times n$ matrices is replaced by the congruence relation. The study of the classical centralizer $\mathcal C_A$ reduces to describing the set of solutions to 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 $n^2$ quadratic equations for $n^2$ unknowns. The complexity of this problem is the reason why we still have no description of the congruent centralizer $C_J^*$ even for the simplest case of the Jordan block $J=J_n(0)$ with zero 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 $n=2,3,4,5$.