Geodesic
Reachability of condensed forms via unitary congruence transformations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1339-1343 
M. Ghasemi Kamalvand; Kh. D. Ikramov. Reachability of condensed forms via unitary congruence transformations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1339-1343. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a0/
@article{ZVMMF_2008_48_8_a0,
     author = {M. Ghasemi Kamalvand and Kh. D. Ikramov},
     title = {Reachability of condensed forms via unitary congruence transformations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1339--1343},
     year = {2008},
     volume = {48},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a0/}
}
TY  - JOUR
AU  - M. Ghasemi Kamalvand
AU  - Kh. D. Ikramov
TI  - Reachability of condensed forms via unitary congruence transformations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 1339
EP  - 1343
VL  - 48
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a0/
LA  - ru
ID  - ZVMMF_2008_48_8_a0
ER  - 
%0 Journal Article
%A M. Ghasemi Kamalvand
%A Kh. D. Ikramov
%T Reachability of condensed forms via unitary congruence transformations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 1339-1343
%V 48
%N 8
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a0/
%G ru
%F ZVMMF_2008_48_8_a0

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

There are several well-known facts about unitary similarity transformations of complex $n$-by-$n$ matrices: every matrix of order $n=3$ can be brought to tridiagonal form by a unitary similarity transformation; if $n\ge5$, then there exist matrices that cannot be brought to tridiagonal form by a unitary similarity transformation; for any fixed set of positions (pattern) $S$ whose cardinality exceeds $n(n-1)/2$, there exists an $n$-by-$n$ matrix $A$ such that none of the matrices that are unitarily similar to $A$ can have zeros in all of the positions in $S$. It is shown that analogous facts are valid if unitary similarity transformations are replaced by unitary congruence ones.

[1] Longstaff W. E., “On tridiagonalization of matrices”, Linear Algebra Appl., 109 (1988), 153–163 | DOI | MR | Zbl

[2] Fong C. K., Wu P. Y., “Band-diagonal operators”, Linear Algebra Appl., 248 (1996), 185–204 | DOI | MR | Zbl

[3] Pati V., “Unitary tridiagonalization in $M_4(\mathbb C)$”, Proc. Indian Acad. Sci. (Math. Sci.), 111 (2001), 381–397 | DOI | MR | Zbl

[4] Davidson K. R., Djoković D. Ž., “Tridiagonal forms in low dimensions”, Linear Algebra Appl., 407 (2005), 169–188 | DOI | MR | Zbl

[5] Djoković D. Ž., MacDonald M. L., “Orthogonal invariants of a matrix of order four and applications”, J. Pure Appl. Algebra, 202 (2005), 259–283 | DOI | MR | Zbl

[6] Djoković D. Ž., Johnson C. R., “Unitarily achievable zero patterns and traces of words in $A$ and $A^*$”, Linear Algebra Appl., 421 (2007), 63–68 | DOI | MR | Zbl

[7] Fassbender H., Ikramov Kh. D., “Some observations on the Youla form and conjugate normal matrices”, Linear Algebra Appl., 422 (2007), 29–38 | DOI | MR | Zbl

[8] Milnor Dzh., Uolles A., Differentsialnaya topologiya. Nachalnyi kurs, Mir, M., 1972 | MR | Zbl

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

[10] Bunse-Gerstner A., Stöver R., “On a conjugate gradient-type method for solving complex symmetric linear systems”, Linear Algebra Appl., 287 (1999), 105–123 | DOI | MR | Zbl