On the Dimension of the Congruence Centralizer

Abstract

Let A be a nonsingular complex (n × n) matrix. The congruence centralizer of A is the collection $$\mathcal{L}$$ of matrices X satisfying the relation $$X{\kern 1pt} {\text{*}}AX = A$$ . The dimension of $$\mathcal{L}$$ 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 $${{A}^{ - }}{\text{*}}A$$ , called the cosquare of A, and the matrix set described by the relation $$X = {{A}^{{ - 1}}}X{\kern 1pt} {\text{*}}A$$ . This dimensional formula is the complex analog of the classical result of A. Voss, which refers to another type of involution in $${{M}_{n}}({\mathbf{C}})$$ .