The congruence centralizer of a block diagonal matrix

Let a complex matrix $A$ be the direct sum of its square submatrices $B$ and $C$ that have no common eigenvalues. Then every matrix $X$ belonging to the centralizer of $A$ has the same block diagonal form as the matrix $A$ itself. In this paper, we discuss how the conditions on the submatrices $B$ and $C$ should be modified to make valid an analogous statement about the congruence centralizer of $A$, which is the set of matrices $X$ such that $X^*AX=A$. We also consider the question whether the matrices in the congruence centralizer are block diagonal if $A$ is a block antidiagonal matrix.