On the dimension of the congruence centralizer
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 493 (2020), pp. 18-20
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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.
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/
@article{DANMA_2020_493_a3,
author = {Kh. D. Ikramov},
title = {On the dimension of the congruence centralizer},
journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
pages = {18--20},
year = {2020},
volume = {493},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/DANMA_2020_493_a3/}
}
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