General algebra and linear transformations preserving matrix invariants

A. È. Guterman ; A. V. Mikhalev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 1, pp. 83-101.

Résumé

The interrelations between the theory of linear transformations preserving matrix invariants and different branches of mathematics are surveyed here. The preferences are given for those methods and motivations to study these transformations that arise from general algebra.

Bibliographie

[1] Artin E., Geometricheskaya algebra, Nauka, M., 1969 | MR | Zbl

[2] Gelfand I. M., Retakh V. S., “Determinanty matrits nad nekommutativnymi koltsami”, Funktsionalnyi analiz i ego prilozheniya, 25 (1991), 91–102 | MR

[3] Guterman A. E., Kreines E. M., Mikhalev A. V., “Rezultaty frobeniusovskogo tipa dlya matrits nad telami”, Trudy pyatykh matematicheskikh chtenii MGSU, 1997, 119–133

[4] Dynkin E. B., “Maksimalnye podgruppy klassicheskikh grupp”, Trudy MMO, 1, 1952, 39–166 | MR | Zbl

[5] Banning R., Mathieu M., “Commutativity preserving mappings on semiprime rings”, Comm. Algebra, 25 (1997), 247–265 | DOI | MR | Zbl

[6] Beasley L., “Linear operators on matrices: The invariance of rank-$k$ matrices”, Linear Algebra Appl., 107 (1988), 161–167 | DOI | MR | Zbl

[7] Beidar K. I., “On functional identities and commuting additive mappings”, Comm. Algebra, 26 (1998), 1819–1850 | DOI | MR | Zbl

[8] Beidar K. I., Brešar M., Chebotar M. A., “Jordan isomorphisms of triangular matrix algebras over a connected commutative ring” (to appear)

[9] Belitskii G. R., Lyubich Yu. I., Matrix Norms and their Applications, Oper. Theory Adv. Appl., 36, Birkhäuser, Boston, 1988 | MR | Zbl

[10] Botta P., Pierce S., Watkins W., “Linear transformations that preserve the nilpotent matrices”, Pacific J. Math., 104 (1983), 39–46 | MR | Zbl

[11] Brešar M., “Commuting traces on beadditive mappings, commutativity preserving mappings, and Lie mappings”, Trans. Amer. Math. Soc., 335 (1993), 525–546 | DOI | MR | Zbl

[12] Brešar M., “Functional identities: A survey”, Contemporary Math. (to appear)

[13] Choi M. D., Jafarian A. A., Radjavi H., “Linear maps preserving commutativity”, Linear Algebra Appl., 87 (1987), 227–242 | DOI | MR

[14] Dedekind R., Gesammelte Mathematische Werke, V. II, Chelsea, New York, 1969

[15] Dieudonné J., “Les déterminants sur un corps non commutatif”, Bull. Soc. Math. Fr., 71 (1943), 27–45 | MR | Zbl

[16] Dieudonné J., “Sur une généralisation du groupe orthogonal à quatre variables”, Arch. Math., 1 (1949), 282–287 | DOI | MR | Zbl

[17] Dixon J. D., “Rigid embedding of simple groups in the general linear group”, Canad. J. Math., 29 (1977), 384–391 | DOI | MR | Zbl

[18] Doković D. Z., Li C. K., “Overgroups of some classical linear groups with applications to linear preserver problems”, Linear Algebra Appl., 197–198 (1994), 31–62 | DOI | MR

[19] Draxl P. K., Skew Fields, London Mathematical Society Lecture Note Series, 81, 1982 | MR

[20] Formanek E., Sibley D., “The group determinant determines the group”, Proc. Amer. Math. Soc., 112 (1991), 649–656 | DOI | MR | Zbl

[21] Frobenius G., Teoriya kharakterov i predstavlenii grupp, Gos. nauch. tekhn. izd. Ukrainy, Kharkov, 1937

[22] Gelfand I. M., Retah V. S., “A theory of noncommutative determinants and characteristic functions of graphs”, Funct. Anal. Appl., 26 (1992), 1–20 ; Publ. LACIM, UQAM, Montreal, 14, 1–26 | DOI | MR

[23] Guralnick R. M., “Invertible preservers and algebraic groups”, Linear Algebra Appl., 212–213 (1994), 249–257 | DOI | MR | Zbl

[24] Guralnick R. M., “Invertible preservers and algebraic groups. II. Preservers of similarity invariants and overgroups of $PSL_n(\mathbf F)$”, Linear and Multilinear Algebra, 43 (1997), 221–255 | DOI | MR | Zbl

[25] Guralnik R. M., Li C.-K., “Invertible preservers and algebraic groups. II: Preservers of unitary similarity (congruence) invariants and overgroups of some unitary subgroups”, Linear and Multilinear Algebra, 43 (1997), 257–282 | DOI | MR

[26] Guterman A., “Frobenius type theorems in the noncommutative case”, Linear and Multilinear Algebra, 48:4 (2001), 293–312 | DOI | MR

[27] Guterman A., “Linear preservers for Drazin star partial order”, Comm. Algebra, 29:9 (2001), 3905–3917 | DOI | MR | Zbl

[28] Guterman A., “Linear preservers for matrix inequalities and partial orderings”, Linear Algebra Appl., 331:1–3 (2001), 75–87 | DOI | MR | Zbl

[29] Guterman A., “Singularity preservers over local domains”, Journal of Mathematical Sciences, 102:6 (2000), 4591–4597 | MR | Zbl

[30] Guterman A., Li C.-K., Šemrl P., “Some general techniques on linear preserver problems”, Linear Algebra Appl., 315 (2000), 61–81 | DOI | MR | Zbl

[31] Guterman A. E., Mikhalev A. V., “Frobenius Type Theorems”, Proceedings of Workshop on General Algebra and Discrete Mathematics, 1998, Shaker-Verlag Aachen, Germany, Potsdam, 1999, 102–112

[32] Guterman A., Mikhalev A. V., “On determinant preservers over noncommutative Principal Ideal Domains”, Lie Algebras, Rings, and Related Topics, Springer-Verlag, Hong Kong, 2000, 49–60 | MR | Zbl

[33] Hiai F., “Similarity preserving linear maps on matrices”, Linear Algebra Appl., 97 (1987), 127–139 | DOI | MR | Zbl

[34] “Hoehnke H.-J.”, Math. Nahr., 1967, no. 34, 229–255 | MR | Zbl

[35] Horn R., Li C.-K., Tsing N. K., “Linear operators preserving certain equivalence relations on matrices”, SIAM J. Matrix Analysis Appl., 12 (1991), 195–204 | DOI | MR | Zbl

[36] Hua L. K., “A theorem on matrices over a sfield and its applications”, J. Chinese Math. Soc. N. S., 1 (1951), 110–163 | MR

[37] Hua L. K., Selected Papers, ed. Halberstam H., Springer-Verlag, New York, Heidelberg, Berlin, 1983 | MR

[38] Jacobson N., Finite-Dimensional Division Algebras over the Fields, Springer-Verlag, New York, Heidelberg, Berlin, 1996 | MR | Zbl

[39] Jacobson N., “Generic norm of an algebra”, Osaca J. Math., 15 (1953), 25–53 | MR

[40] James D. G., “On the automorphisms of $\det(x_{ij})$”, Math. Chronicle, 9 (1980), 35–40 | MR | Zbl

[41] Jensen C. U., Lenzing H., Model Theoretic Algebra with particular emphasis on fields, rings, modules, Algebra, Logic and Applications, 2, Gordon and Breach Science Publishers, New York, 1994 | MR

[42] Johnson K. W., “Latin square determinants, II”, Discrete Mathematics, 105 (1992), 111–130 | DOI | MR | Zbl

[43] Kovacs A., “Trace preserving linear transformations on matrix algebras”, Linear and Multilinear Algebra, 4 (1976/1977), 243–250 | DOI | MR

[44] Lee T.-K., Lee T.-C., “Commuting additive mappings in semiprime rings”, Bull. Inst. Math. Acad. Sinica, 24 (1996), 259–268 | MR | Zbl

[45] Li C.-K., “Norms, isometries and isometry groups”, Amer. Math. Monthly, 107 (2000), 334–340 | DOI | MR | Zbl

[46] Li C.-K., Rodman L., Tsing N. K., “Linear operators preserving certain equivalence relations originating from system theory”, Linear Algebra Appl., 161–165 (1992), 165–225 | DOI | MR | Zbl

[47] Li C.-K., Tsing N. K., “Duality between some linear preserver problems: The inveriance of the $C$-numerical range, the $C$-numerical radius and certain matrix sets”, Linear and Multilinear Algebra, 23 (1988), 353–362 | DOI | MR | Zbl

[48] Li C.-K., Tsing N. K., “Duality between some linear preserver problems. II: Isometries with respect to $c$-spectral norms and matrices with fixed singular values”, Linear Algebra Appl., 110 (1988), 181–212 | MR | Zbl

[49] Li C.-K., Tsing N. K., “Linear preserver problems: A brief introduction and some special techniques”, Linear Algebra Appl., 162–164 (1992), 217–235 | DOI | MR | Zbl

[50] Loewy R., Radwan N., “Spaces of symmetric matrices of bounded rank”, Linear Algebra Appl., Second Conference of the International Linear Algebra Society (ILAS) (Lisbon, 1992), 197–198, 1994, 189–215 | MR | Zbl

[51] Man W. Y., “The invariance of $C$-numerical range, $C$-numerical radius and their dual problems”, Linear and Multilinear Algebra, 30 (1991), 117–128 | DOI | MR | Zbl

[52] Marcus M., “Linear transformations on matrices”, J. Res. Nat. Bur. Stds., 75B (1971), 107–113 | MR | Zbl

[53] Marcus M., May F., “On a theorem of I. Schur concerning matrix transformations”, Arch. Math., 11 (1960), 27–30 | MR

[54] Marcus M., Moyls B., “Linear transformations on algebras of matrices”, Canad. J. Math., 11 (1959), 61–66 | DOI | MR | Zbl

[55] Marcus M., Moyls B., “Transformations on tensor product spaces”, Pacific J. Math., 9 (1959), 1215–1221 | MR | Zbl

[56] Marcus M., Purves R., “Linear transformations on algebras of matrices. II: The invariance of the elementary symmetric functions”, Canad. J. Math., 11 (1959), 383–396 | DOI | MR | Zbl

[57] McDonald B., “$R$-linear endomorphisms of $(R)_n$ preserving invariants”, AMS Memoirs, 287, 1983, 46 | MR

[58] Mikhalev A. V., “Isomorphisms and anti-isomorphisms of endomorphism rings of modules”, First International Tainan–Moscow Algebra Workshop, Walter de Gruyter, Berlin, New York, 1995, 69–122 | MR

[59] Milnor J., Introduction to Algebraic $K$-Theory, Prinston University Press and University of Tokyo Press, 1971 | MR

[60] Omladič M., Šemrl P., “Preserving Diagonalisability”, Linear Algebra Appl., 285 (1998), 165–179 | DOI | MR | Zbl

[61] Pierce S., “Discriminant preserving linear maps”, Linear and Multilinear Algebra, 8 (1979), 101–114 | DOI | MR | Zbl

[62] Pierce S. and others., “A survey of linear preserver problems”, Linear and Multilinear Algebra, 33 (1992), 1–119 | Zbl

[63] Platonov V. P., Doković D. Z., “Linear preserver problems and algebraic groups”, Math. Ann., 303 (1995), 165–184 | DOI | MR | Zbl

[64] Robinson A., Introduction to Model Theory and to the Metamathematics of Algebra, North-Holland Publishing Company, Amsterdam, 1963 | MR | Zbl

[65] Rosenberg J., Algebraic $K$-Theory and its Applications, Springer-Verlag, New York, Berlin, 1994 | MR

[66] Schur I., “Einige Bemerkungen zur Determinantentheorie”, Sitzungsberichte Akad. Berlin, 1925 (1925), 454–463 | Zbl

[67] Wan Z.-X., Geometry of Matrices, World Scientific, Singapore, New Jersey, London, Hong Kong, 1996 | MR

[68] Waterhouse W. C., “Automorphisms of $\det(x_{ij})$: The group scheme approach”, Adv. Math., 65 (1987), 171–203 | DOI | MR | Zbl

[69] Waterhouse W. C., Introduction to Affine Group Schemes, Graduate Text in Math., 66, Springer-Verlag, New York, 1979 | MR | Zbl

[70] Waterhouse W. C., “Invertibility of linear maps preserving matrix invariants”, Linear and Multilinear Algebra, 13 (1983), 105–113 | DOI | MR | Zbl

[71] Waterhouse W. C., “Linear maps preserving reduced norms”, Linear Alg. Appl., 43 (1982), 197–200 | DOI | MR | Zbl

[72] Watkins W., “Linear maps that preserve commuting pairs of matrices”, Linear Algebra Appl., 14 (1976), 29–35 | DOI | MR | Zbl

[73] Westwick R., “Spaces of matrices of fixed rank”, Linear and Multilinear Algebra, 20 (1987), 171–174 | DOI | MR | Zbl

[74] Westwick R., “Transformations on tensor spaces”, Pacific J. of Math., 23:3 (1967), 613–620 | MR | Zbl

[75] Wong W. J., “Maps on simple algebras preserving zero products. I: The associative case”, Pacific J. Math., 89 (1980), 229–247 | MR | Zbl

[76] Wong W. J., “Maps on simple algebras preserving zero products. II: Lie algebras of linear type”, Pacific J. Math., 92 (1981), 469–487 | MR

[77] Wong W. J., “Maps on spaces of linear transformations”, Math. Chronicle, 16 (1987), 15–24 | MR | Zbl

[78] Wong W. J., “Rank $1$ preserving maps on linear transformations over noncommutative local rings”, J. Algebra, 113 (1988), 263–293 | DOI | MR | Zbl