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 
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/
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     title = {Verifying unitary congruence of coninvolutions, skew-coninvolutions, and connilpotent matrices of index two},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] R. Khorn, Ch. Dzhonson, Matrichnyi analiz, Mir, M., 1989 | MR

[2] C. Pearcy, “A complete set of unitary invariants for operators generating finite $W^*$-algebras of type. I”, Pacific J. Math., 12 (1962), 1405–1416 | DOI | MR | Zbl

[3] Y. Hong, R. A. Horn, “A characterization of unitary congruence”, Linear and Multilinear Algebra, 25 (1989), 105–119 | DOI | MR | Zbl

[4] A. George, Kh. D. Ikramov, “Unitary similarity of matrices with quadratic minimal polynomials”, Linear Algebra Appl., 349 (2002), 11–16 | DOI | MR | Zbl

[5] Yu. A. Alpin, Kh. D. Ikramov, “Ob unitarnom podobii algebr, porozhdaemykh parami ortoproektorov”, Zap. nauchn. semin. POMI, 323, 2005, 5–14 | MR | Zbl

[6] R. A. Horn, D. I. Merino, “A real-coninvolutory analog of the polar decomposition”, Linear Algebra Appl., 190 (1993), 209–227 | DOI | MR | Zbl

[7] M. N. M. Abara, D. I. Merino, A. T. Paras, “Skew-coninvolutory matrices”, Linear Algebra Appl., 426 (2007), 540–557 | DOI | MR | Zbl

[8] Kh. D. Ikramov, Kh. Fassbender, “Kvadratichno-normalnye i kongruentno-normalnye matritsy”, Zap. nauchn. semin. POMI, 367, 2009, 45–66 | MR