Geodesic
Conjugate-normal matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XX, Tome 346 (2007), pp. 21-25

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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. 
@article{ZNSL_2007_346_a1,
     author = {M. Ghasemi Kamalvand and Kh. D. Ikramov},
     title = {Conjugate-normal matrices},
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     publisher = {mathdoc},
     volume = {346},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_346_a1/}
}
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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/