Geodesic
On the Unitary Similarity of Matrix Families
Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 815-826.

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

The classical Specht criterion for the unitary similarity between two complex $(n\times n)$ matrices is extended to the unitary similarity between two normal matrix sets of cardinality $m$. This property means that the algebra generated by a set is closed with respect to the conjugate transpose operation. Similar to the well-known result of Pearcy that supplements Specht"s theorem, the proposed extension can be made a finite criterion. The complexity of this criterion depends on n as well as the length l of the algebras under analysis. For a pair of matrices, this complexity can be significantly lower than that of the Specht–Pearcy criterion. 
@article{MZM_2003_74_6_a1,
     author = {Yu. A. Alpin and Kh. D. Ikramov},
     title = {On the {Unitary} {Similarity} of {Matrix} {Families}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {815--826},
     publisher = {mathdoc},
     volume = {74},
     number = {6},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a1/}
}
TY  - JOUR
AU  - Yu. A. Alpin
AU  - Kh. D. Ikramov
TI  - On the Unitary Similarity of Matrix Families
JO  - Matematičeskie zametki
PY  - 2003
SP  - 815
EP  - 826
VL  - 74
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a1/
LA  - ru
ID  - MZM_2003_74_6_a1
ER  - 
%0 Journal Article
%A Yu. A. Alpin
%A Kh. D. Ikramov
%T On the Unitary Similarity of Matrix Families
%J Matematičeskie zametki
%D 2003
%P 815-826
%V 74
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a1/
%G ru
%F MZM_2003_74_6_a1
Yu. A. Alpin; Kh. D. Ikramov. On the Unitary Similarity of Matrix Families. Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 815-826. http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a1/

[1] Shapiro H., “A survey of canonical forms and invariants for unitary similarity”, Linear Algebra and Appl., 147 (1991), 101–167 | DOI | MR | Zbl

[2] Kaluzhnin L. A., Khavidi Kh., “Geometricheskaya teoriya unitarnoi ekvivalentnosti matrits”, Dokl. RAN, 169:5 (1996), 1009–1012

[3] Van der Varden B. L., Algebra, Nauka, M., 1979

[4] Specht W., “Zur Theorie der Matrizen, II”, Jahresber. Deutsch. Math.-Verein, 50 (1940), 19–23 | MR | Zbl

[5] Khorn R., Dzhonson Ch., Matrichnyi analiz, Mir, M., 1989

[6] Vinberg E. B., Kurs algebry, Faktorial, M., 1999

[7] Veil G., Klassicheskie gruppy, ikh invarianty i predstavleniya, IL, M., 1947

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

[9] Voevodin V. V., Kuznetsov Yu. A., Matritsy i vychisleniya, Nauka, M., 1984

[10] Alpin Yu. A., Ikramov Kh. D., “Ob unitarnom podobii semeistv matrits”, Dokl. RAN, 384:2 (2002), 151–152 | MR

[11] Freedman A., Gupta R. N., Guralnik R. M., “Shirshov's theorem and representations of semigroups”, Pacific J. Math., 181:3 (1997), 159–176 | MR

[12] Pappacena C., “An upper bound for the length of a finite-dimensional algebra”, J. Algebra, 197 (1997), 535–545 | DOI | MR | Zbl

[13] Gaines F. T., Laffey T. J., Shapiro H. M., “Pairs of matrices with quadratic minimal polynomials”, Linear Algebra and Appl., 52/53 (1983), 289–292 | MR | Zbl