Geodesic
Commutativity of matrices up to a matrix factor
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXII, Tome 482 (2019), pp. 151-168 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The matrix relation $ AB = CBA $ is investigated. An explicit description of the space of matrices $B$ satisfying this relation is obtained for an arbitrary fixed matrix $C$ and a diagonalizable matrix $A$. The connection between this space and the family of right annihilators of the matrices $A- \lambda C $, where $ \lambda $ ranges over the set of eigenvalues of the matrix $A$, is studied. In the case where $ AB = CBA $, $ AC = CA $, $ BC = CB $, a canonical form for $ A, B, C$, generalizing Thompson's result for invertible $ A, B, C,$ is introduced. Also bounds for the length of pairs of matrices $ \{A, B \} $ of the form indicated are provided. 
@article{ZNSL_2019_482_a10,
     author = {N. A. Kolegov and O. V. Markova},
     title = {Commutativity of matrices up to a matrix factor},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {151--168},
     year = {2019},
     volume = {482},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_482_a10/}
}
TY  - JOUR
AU  - N. A. Kolegov
AU  - O. V. Markova
TI  - Commutativity of matrices up to a matrix factor
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2019
SP  - 151
EP  - 168
VL  - 482
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2019_482_a10/
LA  - ru
ID  - ZNSL_2019_482_a10
ER  - 
%0 Journal Article
%A N. A. Kolegov
%A O. V. Markova
%T Commutativity of matrices up to a matrix factor
%J Zapiski Nauchnykh Seminarov POMI
%D 2019
%P 151-168
%V 482
%U http://geodesic.mathdoc.fr/item/ZNSL_2019_482_a10/
%G ru
%F ZNSL_2019_482_a10
N. A. Kolegov; O. V. Markova. Commutativity of matrices up to a matrix factor. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXII, Tome 482 (2019), pp. 151-168. http://geodesic.mathdoc.fr/item/ZNSL_2019_482_a10/

[1] A. Alahmadi, S. P. Glasby, C. E. Praeger, “On the dimension of twisted centralizer codes”, Finite Fields Appl., 48 (2017), 43–59 | DOI | MR | Zbl

[2] J. A. Brooke, P. Busch, D. B. Pearson, “Commutativity up to a factor of bounded operators in complex Hilbert space”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458:2017 (2002), 109–118 | DOI | MR | Zbl

[3] N. Chriss, V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, Boston–Basel–Berlin, 1997 | MR | Zbl

[4] S. M. Chuiko, “O reshenii bilineinogo matrichnogo uravneniya”, Chebyshevskii sb., 17:2 (2016), 196–205 | DOI | MR | Zbl

[5] G. Dolinar, A. Guterman, B. Kuzma, O. Markova, “Double centralizing theorem with respect to $q$-commutativity relation”, J. Algebra Appl., 18:1 (2019), 1950003 | DOI | MR | Zbl

[6] G. Dolinar, A. Guterman, B. Kuzma, O. Markova, “Extremal generalized centralizers in matrix algebras”, Comm. Algebra, 46:7 (2018), 3147–3154 | DOI | MR | Zbl

[7] A. Guterman, T. Laffey, O. Markova, H. Šmigoc, “A resolution of Paz's conjecture in the presence of a nonderogatory matrix”, Linear Algebra Appl., 543 (2018), 234–250 | DOI | MR | Zbl

[8] A. E. Guterman, O. V. Markova, “Commutative matrix subalgebras and length function”, Linear Algebra Appl., 430 (2009), 1790–1805 | DOI | MR | Zbl

[9] A. E. Guterman, O. V. Markova, “Problema realizuemosti znachenii dliny dlya pary kvazi-kommutiruyuschikh matrits”, Zap. nauchn. semin. POMI, 439, 2015, 59–73

[10] A. E. Guterman, O. V. Markova, V. Mehrmann, “Lengths of quasi-commutative pairs of matrices”, Linear Algebra Appl., 498 (2016), 450–470 | DOI | MR | Zbl

[11] A. E. Guterman, O. V. Markova, V. Mehrmann, “Length realizability for pairs of quasi-commuting matrices”, Linear Algebra Appl., 568 (2019), 135–154 | DOI | MR | Zbl

[12] C. Kassel, Quantum Groups, Graduate Texts Math., 155, Springer-Verlag, New York, 1995 | DOI | MR | Zbl

[13] N. A. Kolegov, O. V. Markova, “Sistemy porozhdayuschikh matrichnykh algebr intsidentnosti nad konechnymi polyami”, Zap. nauchn. semin. POMI, 472, 2018, 120–144

[14] W. E. Longstaff, P. Rosenthal, “On the lengths of irreducible pairs of complex matrices”, Proc. Amer. Math. Soc., 139:11 (2011), 3769–3777 | DOI | MR | Zbl

[15] A. I. Maltsev, Osnovy lineinoi algebry, Nauka, M., 1970 | MR

[16] Yu. I. Manin, Quantum Groups and Non-Commutative Geometry, CRM, Montréal, 1988 | MR | Zbl

[17] O. V. Markova, “Kharakterizatsiya kommutativnykh matrichnykh podalgebr maksimalnoi dliny nad proizvolnym polem”, Vestn. Mosk. univ. Ser. 1. Matematika. Mekhanika, 5 (2009), 53–55 | Zbl

[18] H. Neudecker, “A note on Kronecker matrix products and matrix equation systems”, SIAM J. Appl. Math., 18:3 (1969), 603–606 | DOI | MR

[19] H. Neudecker, “Some theorems on matrix differentiation with special reference to kronecker matrix products”, J. Amer. Statist. Assoc., 64:327 (1969), 953–963 | DOI | Zbl

[20] A. Paz, “An application of the Cayley–Hamilton theorem to matrix polynomials in several variables”, Linear Multilinear Algebra, 15 (1984), 161–170 | DOI | MR | Zbl

[21] H. Shapiro, “Commutators which commute with one factor”, Pacific J. Math., 1997, Special Issue: “Olga Taussky-Todd: in memoriam”, 323–336 | DOI | MR

[22] C. Song, G. Chen, L. Zhao, “Iterative solutions to coupled Sylvester-transpose matrix equations”, Appl. Math. Model., 35 (2011), 4675–4683 | DOI | MR | Zbl

[23] R. C. Thompson, “Multiplicative matrix commutators commuting with both factors”, J. Math. Annl. Appl., 18 (1967), 315–335 | DOI | MR | Zbl

[24] R. C. Thompson, “Some matrix factorization theorems. I”, Pacific. J. Math., 33:3 (1970), 763–810 | DOI | MR | Zbl

[25] R. C. Thompson, “Some matrix factorization theorems. II”, Pacific. J. Math., 33:3 (1970), 811–822 | DOI | MR | Zbl

[26] G. Veil, Teoriya grupp i kvantovaya mekhanika, Nauka, M., 1986