Geodesic
Monotone matrix transformations defined
Sbornik. Mathematics, Tome 198 (2007) no. 1, pp. 1-16 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Bijective linear transformations of the matrix algebra over an arbitrary field that preserve simultaneous diagonalizability are characterized. This result is used for the characterization of bijective linear transformations monotone with respect to the $\stackrel{\sharp}<$- and $\stackrel{\mathrm{cn}}<$-orders. Bibliography: 28 titles. 
@article{SM_2007_198_1_a0,
     author = {I. I. Bogdanov and A. \`E. Guterman},
     title = {Monotone matrix transformations defined},
     journal = {Sbornik. Mathematics},
     pages = {1--16},
     year = {2007},
     volume = {198},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2007_198_1_a0/}
}
TY  - JOUR
AU  - I. I. Bogdanov
AU  - A. È. Guterman
TI  - Monotone matrix transformations defined
JO  - Sbornik. Mathematics
PY  - 2007
SP  - 1
EP  - 16
VL  - 198
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_2007_198_1_a0/
LA  - en
ID  - SM_2007_198_1_a0
ER  - 
%0 Journal Article
%A I. I. Bogdanov
%A A. È. Guterman
%T Monotone matrix transformations defined
%J Sbornik. Mathematics
%D 2007
%P 1-16
%V 198
%N 1
%U http://geodesic.mathdoc.fr/item/SM_2007_198_1_a0/
%G en
%F SM_2007_198_1_a0
I. I. Bogdanov; A. È. Guterman. Monotone matrix transformations defined. Sbornik. Mathematics, Tome 198 (2007) no. 1, pp. 1-16. http://geodesic.mathdoc.fr/item/SM_2007_198_1_a0/

[1] M. J. Englefield, “The commuting inverses of a square matrix”, Proc. Cambridge Philos. Soc., 62 (1966), 667–671 | DOI | MR | Zbl

[2] S. K. Mitra, “A new class of $g$-inverse of square matrices”, Sankhyā Ser. A, 30 (1968), 323–330 | MR | Zbl

[3] S. K. Mitra, “On group inverses and the sharp order”, Linear Algebra Appl., 92 (1987), 17–37 | DOI | MR | Zbl

[4] P. Robert, “On the group-inverse of a linear transformation”, J. Math. Anal. Appl., 22 (1968), 658–669 | DOI | MR | Zbl

[5] I. Erdelyi, “On the matrix equation $Ax=\lambda Bx$”, J. Math. Anal. Appl., 17 (1967), 117–132 | DOI | MR | Zbl

[6] A. Ben-Israel, T. Greville, Generalized inverses: theory and applications, Wiley, New York, 1974 | MR | Zbl

[7] C. R. Rao, S. K. Mitra, Generalized inverse of matrices and its applications, Wiley, New York, 1971 | MR | Zbl

[8] R. E. Hartwig, “How to partially order regular elements”, Math. Japon., 25:1 (1980), 1–13 | MR | Zbl

[9] K. S. S. Nambooripad, “The natural partial order on a regular semigroup”, Proc. Edinb. Math. Soc. (2), 23 (1980), 249–260 | DOI | MR | Zbl

[10] M. P. Drazin, “Natural structures on semigroups with involution”, Bull. Amer. Math. Soc., 84:1 (1978), 139–141 | DOI | MR | Zbl

[11] S. K. Mitra, “Matrix partial orders through generalized inverses: Unified theory”, Linear Algebra Appl., 148 (1991), 237–263 | DOI | MR | Zbl

[12] S. K. Mitra, “The minus partial order and the shorted matrix”, Linear Algebra Appl., 83 (1986), 1–27 | DOI | MR | Zbl

[13] J. K. Baksalary, F. Pukelsheim, G. P. H. Styan, “Some properties of matrix partial orderings”, Linear Algebra Appl., 119 (1989), 57–85 | DOI | MR | Zbl

[14] J. K. Baksalary, J. Hauke, “A further algebraic version of Cochran's theorem and matrix partial orderings”, Linear Algebra Appl., 127 (1990), 157–169 | DOI | MR | Zbl

[15] S. K. Mitra, R. E. Hartwig, “Partial orders based on outer inverses”, Linear Algebra Appl., 176 (1992), 3–20 | DOI | MR | Zbl

[16] G. Frobenius, “Über die Darstellung der endlichen Gruppen durch lineare Substitutionen”, Sitzungsber. Preuss. Akad. Wiss. Berlin, 1897, 994–1015 ; G. Frobenius, Teoriya kharakterov i predstavlenii grupp, GNTI Ukrainy, Kharkov, 1937, 106–127 | Zbl

[17] A. E. Guterman, A. V. Mikhalev, “Obschaya algebra i lineinye otobrazheniya, sokhranyayuschie matrichnye invarianty”, Fundam. prikl. matem., 9:1 (2003), 83–101 ; A. E. Guterman, A. V. Mikhalev, “General algebra and linear transformations preserving matrix invariants”, J. Math. Sci., 128:6 (2005), 3384–3395 | MR | Zbl | DOI

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

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

[20] J. de Pillis, “Linear transformations which preserve Hermitian and positive semidefinite operators”, Pacific J. Math., 23 (1967), 129–137 | MR | Zbl

[21] A. A. Alieva, A. E. Guterman, “Monotone linear transformations on matrices are invertible”, Comm. Algebra, 33:9 (2005), 3335–3352 | DOI | MR | Zbl

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

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

[24] A. Guterman, “Monotone matrix maps preserve non-maximal rank”, Polynomial identities and combinatorial methods, Lect. Notes Pure Appl. Math., 235, Dekker, New York, 2003, 311–328 | MR | Zbl

[25] M. Omladič P. Šemrl, “Preserving diagonalisability”, Linear Algebra Appl., 285:1–3 (1998), 165–179 | DOI | MR | Zbl

[26] T. S. Motzkin, O. Taussky, “Pairs of matrices with property $L$”, Trans. Amer. Math. Soc., 73 (1952), 108–114 | DOI | MR | Zbl

[27] T. S. Motzkin, O. Taussky, “Pairs of matrices with property $L$. II”, Trans. Amer. Math. Soc., 80 (1955), 387–401 | DOI | MR | Zbl

[28] A. I. Maltsev, Osnovy lineinoi algebry, 3 izd., Nauka, M., 1975 ; A. I. Mal'cev, Foundations of linear algebra, Freeman, San Francisco, 1963 | MR | MR