The congruent centralizer of the Jordan block (Q2405043)

Here, all matrices are \(n\times{n}\) complex. The congruent centralizer \(\mathcal{C}_A^*\) of a matrix \(A\) is defined to be the set of matrices \(X\) satisfying \(X^*AX=A\), in analogy with the classical centralizer using similarity. The structure of the latter is well known, but describing \(\mathcal{C}_A^*\) is equivalent to solving a system of \(n^2\) quadratic equations in \(n^2\) variables (the entries of the matrix \(X\)). This has not been achieved except in a few cases of \(A\), not even for the Jordan block \(J=J_n(0)\) with zeros on the principal diagonal. The author proves certain facts about the structure of the matrices in \(\mathcal{C}_J^*\) for arbitrary \(n\) and gives complete descriptions for the cases \(n=2,3,4\) and \(5\).