Geodesic
Continuous Adjacency Preserving Maps on Real Matrices
Canadian mathematical bulletin, Tome 48 (2005) no. 2, pp. 267-274

Voir la notice de l'article provenant de la source Cambridge University Press

It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $A\,\mapsto \,PAQ\,+\,R$ , possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$ . The result extends previously known theorems where the map was assumed to be also injective.
DOI : 10.4153/CMB-2005-025-6
Mots-clés : 15A03, 15A04, adjacency of matrices, continuous preservers, affine transformations 
Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R. Continuous Adjacency Preserving Maps on Real Matrices. Canadian mathematical bulletin, Tome 48 (2005) no. 2, pp. 267-274. doi: 10.4153/CMB-2005-025-6
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[1] [1] Carter, D. S. and Vogt, A., Collinearity-preserving functions between Desarguesian planes. Mem. Amer. Math. Soc. 27(1980), 1–98. Google Scholar

[2] [2] Fulton, W., Algebraic topology. A first course. Springer-Verlag, New York, 1995. Google Scholar

[3] [3] Hua, L. K., Geometries of matrices I. Generalizations of von Staudt's theorem. Trans. Amer. Math. Soc. 57(1945), 441–481. Google Scholar

[4] [4] Hua, L. K., Geometries of matrices I . Arithmetical construction. Trans. Amer.Math. Soc. 57(1945), 482–490. Google Scholar

[5] [5] Hua, L. K., Geometries of symmetric matrices over the real field, I. C. R. (Doklady) Acad. Sci. URSS 53(1946), 95–97. Google Scholar

[6] [6] Hua, L. K., Geometries of symmetric matrices over the real field, II. C. R. (Doklady) Acad. Sci. URSS 53(1946), 195–196. Google Scholar

[7] [7] Hua, L. K., Geometries of matrices, II. Study of involutions in the geometry of symmetric matrices. Trans. Amer. Math. Soc. 61(1947), 193–228. Google Scholar

[8] [8] Hua, L. K., Geometries of matrices, III. Fundamental theorems in the geometries of symmetric matrices. Trans. Amer. Math. Soc. 61(1947), 229–255. Google Scholar

[9] [9] Hua, L. K., Geometry of symmetric matrices over any field with characteristic other than two. Ann. of Math. 50(1949), 8–31. Google Scholar

[10] [10] Hua, L. K., A theorem on matrices over a sfield and its applications. J. Chinese Math. Soc. 1(1951), 110–163. Google Scholar

[11] [11] Lim, M. H., Rank and tensor rank preservers, Linear and Multilinear Algebra 33(1992), 7–21. Google Scholar

[12] [12] Petek, T. and Šemrl, P., Adjacency preserving maps on matrices and operators. Proc. Roy. Soc. Edinburgh Sect. A. 132(2002), 661–684. Google Scholar

[13] [13] Šemrl, P., On Hua's fundamental theorem of the geometry of rectangular matrices. J. Algebra 248(2002), 366–380. Google Scholar

[14] [14] Šemrl, P., Hua's fundamental theorems of the geometry of matrices and related results. Linear Algebra Appl. 361(2003), 161–179. Google Scholar

[15] [15] Wan, Z., Geometry of matrices. World Scientific, River Edge, NJ, 1996. Google Scholar

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