Geodesic
On tridiagonal conjugate-normal matrices
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 2, pp. 179-185 
Kh. D. Ikramov. On tridiagonal conjugate-normal matrices. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 2, pp. 179-185. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_2_a0/
@article{ZVMMF_2007_47_2_a0,
     author = {Kh. D. Ikramov},
     title = {On tridiagonal conjugate-normal matrices},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {179--185},
     year = {2007},
     volume = {47},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_2_a0/}
}
TY  - JOUR
AU  - Kh. D. Ikramov
TI  - On tridiagonal conjugate-normal matrices
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2007
SP  - 179
EP  - 185
VL  - 47
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_2_a0/
LA  - ru
ID  - ZVMMF_2007_47_2_a0
ER  - 
%0 Journal Article
%A Kh. D. Ikramov
%T On tridiagonal conjugate-normal matrices
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2007
%P 179-185
%V 47
%N 2
%U http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_2_a0/
%G ru
%F ZVMMF_2007_47_2_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

Conjugate-normal matrices play the same role in the theory of unitary congruences as conventional normal matrices do with respect to unitary similarities. Naturally, the properties of both matrix classes are fairly similar up to the distinction between the congruence and similarity. However, in certain respects, conjugate-normal matrices differ substantially from normal ones. Our goal in this paper is to indicate one of such distinctions. It is shown that none of the familiar characterizations of normal matrices having the irreducible tridiagonal form has a natural counterpart in the case of conjugate-normal matrices.

[1] Ikramov Kh. D., “Neravenstvo Shura dlya psevdosobstvennykh znachenii i sopryazhenno-normalnye matritsy”, Vestn. MGU. Ser. 15. Vychisl. matem. i kibernetika, 1997, no. 3, 3–6 | MR

[2] George A., Ikramov Kh. D., Matushkina T. V., Tang W.-P., “On a QR-like algorithm for some structured eigenvalue problems”, SIAM J. Matrix Analys and Appl., 16 (1995), 1107–1126 | DOI | MR | Zbl

[3] Khorn P., Dzhonson Ch., Matrichnyi analiz, Mir, M., 1989 | MR

[4] Wigner E. P., “Normal form of antiunitary operators”, J. Math. Phys., 1 (1960), 409–413 | DOI | MR | Zbl

[5] Vujicić M., Herbut F., Vujicić G., “Canonical forms for matrices under unitary congruence transformations I: conjugate-normal matrices”, SIAM J. Appl. Math., 23 (1972), 225–238 | DOI | MR | Zbl

[6] Ikramov Kh. D., “O psevdosobstvennykh znacheniyakh i singulyarnykh chislakh kompleksnoi kvadratnoi matritsy”, Zap. nauchn. seminarov POMI, SPb., 2006, 111–120 | MR | Zbl