Unitary congruence to a~conjugate-normal matrix
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 133-137
Voir la notice de l'article provenant de la source Math-Net.Ru
A matrix $A\in M_n(\mathbb C)$ is said to be conjugate-normal if $AA^*=\overline{A^*A}.$ The following proposition (which is the congruence analog of a recent result of T. G. Gerasimova) is proved: A matrix $B\in M_n(\mathbb C)$ is unitarily congruent to a conjugate-normal matrix $A$ if and only if
$$
\mathrm{tr}[(\bar AA)^i]=\mathrm{tr}[(\bar BB)^i],\qquad i=1,\dots,n,
$$
and
$$
\|A\|_F=\|B\|_F.
$$
This proposition dramatically reduces the amount of computational work for verifying unitary congruence as compared to the case of general matrices $A$ and $B$.
@article{ZNSL_2012_405_a10,
author = {Kh. D. Ikramov},
title = {Unitary congruence to a~conjugate-normal matrix},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {133--137},
publisher = {mathdoc},
volume = {405},
year = {2012},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a10/}
}
Kh. D. Ikramov. Unitary congruence to a~conjugate-normal matrix. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 133-137. http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a10/