Geodesic
Unitary congruence to a conjugate-normal matrix
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 133-137 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $B$. 
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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/

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