Geodesic
Conjugate-normal matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XX, Tome 346 (2007), pp. 21-25 
M. Ghasemi Kamalvand; Kh. D. Ikramov. Conjugate-normal matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XX, Tome 346 (2007), pp. 21-25. http://geodesic.mathdoc.fr/item/ZNSL_2007_346_a1/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In studying the reduction of a complex $n\times n$ matrix $A$ to its Hessenberg form by the Arnoldi algorithm, T. Huckle discovered that an irreducible Hessenberg normal matrix with a normal leading principal $m\times m$ submatrix, where $1, actually is tridiagonal. We prove a similar assertion for the conjugate-normal matrices, which play the same role in the theory of unitary congruences as the conventional normal matrices in the theory of unitary similarities. This fact is stated as a purely matrix-theoretic theorem, without any reference to Arnoldi-like algorithms.

[1] T. Huckle, “The Arnoldi method for normal matrices”, SIAM J. Matrix Anal. Appl., 15:2 (1994), 479–489 | DOI | MR | Zbl

[2] Kh. D. Ikramov, L. Elzner, “O normalnykh matritsakh s normalnymi glavnymi podmatritsami”, Zap. nauchn. semin. POMI, 229, 1995, 63–94 | MR