Geodesic
On the dimension of the congruence centralizer
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 493 (2020), pp. 18-20 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $M_n(\mathbf{C})$.
Keywords: $^*$-congruence, congruence centralizer, cosquare, canonical form with respect to congruences. 
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Kh. D. Ikramov. On the dimension of the congruence centralizer. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 493 (2020), pp. 18-20. http://geodesic.mathdoc.fr/item/DANMA_2020_493_a3/

[1] Voss A., “Ueber die cogredienten Transformationen einer bilinearer Form in sich selbst”, Abh. bayer. Akad. Wiss. II, 17 (1892), 233–356

[2] De Terán F., Dopico F.M., “The equation $XA + AX * = 0$ and the dimension of *-congruence orbits”, Electronic J. Linear Algebra, 22 (2011), 448–465 | MR | Zbl

[3] Gantmakher F.R., Teoriya matrits, Nauka, M., 1966 | MR