Skip to main content
SpringerLink
Log in

Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution

  • ORIGINAL ARTICLES
  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let D be any division ring with an involution, ℋ n (D) be the space of all n × n hermitian matrices over D. Two hermitian matrices A and B are said to be adjacent if rank(A − B) = 1. It is proved that if ϕ is a bijective map from ℋ n (D)(n ≥ 2) to itself such that ϕ preserves the adjacency, then ϕ −1 also preserves the adjacency. Moreover, if ℋ n (D ≠ \({\fancyscript S}\) 3(\({\fancyscript F}\) 2), then ϕ preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe–Xian is answered for geometry of symmetric and hermitian matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Wan, Z. X.: Geometry of Matrices: In Memory of Professor L K Hua (1910–1985), World Scientific, Singapore, 1996

  2. Hua, L. K.: Geometries of symmetric matrices any field with characteristic other than two. Ann. of Math., 50, 8–31 (1949)

    Article  MathSciNet  Google Scholar 

  3. Wan, Z. X.: Geometries of symmetric matrices and its applications I. Algebra Colloquium, 1(2), 97–120 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Wan, Z. X.: Geometries of symmetric matrices and its applications II. Algebra Colloquium, 1(3), 201–224 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Gao, J. F., Wan, Z. X., Feng, R. Q., Wang, D. J.: Geometry of symmetric matrices and its applications III. Algebra Colloquium, 3(2), 135–146 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Wan, Z. X.: Geometry of Hamiltonian matrices and Its applications I. Algebra Colloquium, 2(2), 167–192 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Wan, Z. X.: Geometry of Hamiltonian matrices and Its applications II. Algebra Colloquium, 3(2), 107–116 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Huang, L. P., Wan, Z. X.: Geometry of 2 × 2 hermitian matrices. Science in China (Series A), 45(8), 1025–1037 (2002)

    Article  MATH  Google Scholar 

  9. Wan, Z. X.: Geometry of Matrices of revisited, Algebra and Combinatorics, An International Congress, ICAC’97, Hong Kong, Edited by K. –P. Shum et al., Springer, Singapore, 477–486, 1999

  10. Huang, W. L., Höfer, R., Wan, Z. X.: Adjacency preserving mappings of symmetric and hermitian matrices. Aequations Math., 67(1–2), 132–139 (2004)

    Article  MATH  Google Scholar 

  11. Hua, L. K., Wan, Z. X.: Classical Groups, Shanghai Science and Technology Press, Shanghai, 1963 (in Chinese)

  12. Huang, L. P.: The quaternion matrix equation ∑A i XB i = E. Acta Mathematica Sinica, English Series, 14(1), 91–98 (1998)

    MATH  Google Scholar 

  13. Fošner, A., Sěmrl, P.: Additive maps on matrix algebras preserving invertibility or singularity. Acta Mathematica Sinica, English Series, 21(4), 681–684 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hou, J. C., Zhang, X. L.: Additive maps preserving similarity of operators on Banach spaces. Acta Mathematica Sinica, English Series, 22(1), 179–186 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Changsha University of Science & Technology, Changsha 410076, P. R. China

    Li Ping Huang

Authors
  1. Li Ping Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Ping Huang.

Additional information

This work is supported by the National Natural Science Foundation of China Grant #10271021

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, L.P. Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution. Acta Math Sinica 23, 95–102 (2007). https://doi.org/10.1007/s10114-005-0770-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-005-0770-7

Keywords

MR (2000) Subject Classification

Search

Navigation