The congruent centralizer of the Horn-Sergeichuk matrix (Q2405045)

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 only been achieved in a few cases, for instance \(A=I_n\), or \(I_p\oplus(-I_q)\), or \(\begin{pmatrix} 0&I_m\\ -I_m&0 \end{pmatrix}\), in which case \(\mathcal{C}_A^*\) is the group of unitary matrices, or the group of pseudo-unitary matrices of type \((p,q)\), or the symplectic group of order \(n=2m\), respectively. In this paper, the author studies \(\mathcal{C}_\Delta^*\), where \(\Delta=\begin{pmatrix} &&&1\\ &&\dots&i\\ &1&\dots&\\ 1&i&& \end{pmatrix}\), which is one of the three blocks in the Horn-Sergeichuk canonical form. The main result of the paper is the theorem: up to special diagonal factors all matrices in \(\mathcal{C}_\Delta^*\) are upper triangular and Toeplitz.