On the dimension of the congruence centralizer

Let $A$ be a nonsingular complex $(n\times n)$ matrix. The congruence centralizer of $A$ is the collection $\mathscr{L}$ of matrices $X$ satisfying the relation $X^*AX=A$. The dimension of $\mathscr{L}$ as a real variety in the matrix space $M_n(\mathbf{C})$ is shown to be equal to the difference of the real dimensions of the following two sets: the conventional centralizer of the matrix $A^{-*}A$, called the cosquare of $A$, and the matrix set described by the relation $X=A^{-1}X^*A$. This dimensional formula is the complex analog of the classical result of A. Voss, which refers to another type of involution in $M_n(\mathbf{C})$.