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Induced Maps on Matrices over Fields

Author

Listed:
  • Li Yang
  • Xuezhi Ben
  • Ming Zhang
  • Chongguang Cao

Abstract

Suppose that 𝔽 is a field and m, n β‰₯ 3 are integers. Denote by Mmn(𝔽) the set of all m Γ— n matrices over 𝔽 and by Mn(𝔽) the set Mnn(𝔽). Let fij (i ∈ [1, m], j ∈ [1, n]) be functions on 𝔽, where [1, n] stands for the set {1, …, n}. We say that a map f : Mmn(𝔽) β†’ Mmn(𝔽) is induced by {fij} if f is defined by f : [aij] ↦ [fij(aij)]. We say that a map f on Mn(𝔽) preserves similarity if A ~ Bβ‡’f(A) ~ f(B), where A ~ B represents that A and B are similar. A map f on Mn(𝔽) preserving inverses of matrices means f(A)f(Aβˆ’1) = In for every invertible A ∈ Mn(𝔽). In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.

Suggested Citation

  • Li Yang & Xuezhi Ben & Ming Zhang & Chongguang Cao, 2014. "Induced Maps on Matrices over Fields," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:596756
    DOI: 10.1155/2014/596756
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    References listed on IDEAS

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    1. Jinli Xu & Baodong Zheng & Li Yang, 2013. "The Automorphism Group of the Lie Ring of Real Skew-Symmetric Matrices," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, August.
    2. Hongmei Yao & Baodong Zheng, 2012. "Zero Triple Product Determined Matrix Algebras," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Li Yang & Wei Zhang & Jinli Xu, 2014. "Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-8, January.
    4. K. L. Lin & D. P. Yang & J. C. Yao, 1997. "Generalized Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 117-125, January.
    5. Li Yang & Wei Zhang & Jinli Xu, 2014. "Linear Maps on Upper Triangular Matrices Spaces Preserving Idempotent Tensor Products," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    6. Hongmei Yao & Baodong Zheng, 2012. "Zero Triple Product Determined Matrix Algebras," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-18, February.
    7. Jinli Xu & Baodong Zheng & Li Yang, 2013. "The Automorphism Group of the Lie Ring of Real Skew‐Symmetric Matrices," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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