Geodesic
Verifying unitary congruence of coninvolutions, skew-coninvolutions, and connilpotent matrices of index two
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXII, Tome 367 (2009), pp. 27-32

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It is shown that $n\times n$ solutions $A$ and $B$ of the matrix equation $$ X\overline X=\delta I, $$ where $\delta$ is one and the same scalar for both matrices, are unitarily congruent if and only if $$ \operatorname{tr}(A^*A)^k=\operatorname{tr}(B^*B)^k,\qquad k=1,2,\dots,n. $$ Bibl. – 8 titles. 
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     author = {Kh. D. Ikramov},
     title = {Verifying unitary congruence of coninvolutions, skew-coninvolutions, and connilpotent matrices of index two},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {27--32},
     publisher = {mathdoc},
     volume = {367},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_367_a2/}
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Kh. D. Ikramov. Verifying unitary congruence of coninvolutions, skew-coninvolutions, and connilpotent matrices of index two. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXII, Tome 367 (2009), pp. 27-32. http://geodesic.mathdoc.fr/item/ZNSL_2009_367_a2/