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Continuous Adjacency Preserving Maps on Real Matrices

Published online by Cambridge University Press:  20 November 2018

Leiba Rodman ,
Peter Šemrl  and
Ahmed R. Sourour
Leiba Rodman
Affiliation:
Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, U.S.A. e-mail: lxrodm@math.wm.edu
Peter Šemrl
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia e-mail: peter.semrl@fmf.uni-lj.si
Ahmed R. Sourour
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3P4 e-mail: sourour@math.uvic.ca
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Abstract

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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.

Keywords

15A0315A04adjacency of matricescontinuous preserversaffine transformations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 2 , 01 June 2005 , pp. 267 - 274
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Carter, D. S. and Vogt, A., Collinearity-preserving functions between Desarguesian planes. Mem. Amer. Math. Soc. 27(1980), 1–98.Google Scholar
[2] Fulton, W., Algebraic topology. A first course. Springer-Verlag, New York, 1995.Google Scholar
[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] Hua, L. K., Geometries of matrices I 1 . Arithmetical construction. Trans. Amer.Math. Soc. 57(1945), 482–490.Google Scholar
[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] 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] 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] 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] 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] Hua, L. K., A theorem on matrices over a sfield and its applications. J. Chinese Math. Soc. 1(1951), 110–163.Google Scholar
[11] Lim, M. H., Rank and tensor rank preservers, Linear and Multilinear Algebra 33(1992), 7–21.Google Scholar
[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] Šemrl, P., On Hua's fundamental theorem of the geometry of rectangular matrices. J. Algebra 248(2002), 366–380.Google Scholar
[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] Wan, Z., Geometry of matrices. World Scientific, River Edge, NJ, 1996.Google Scholar