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Determinants Preserving Maps on the Spaces of Symmetric Matrices and Skew-Symmetric Matrices

Abstract

Denote by Σn and Qn the set of all n × n symmetric and skew-symmetric matrices over a field \(\mathbb {F}\), respectively, where \(\text {char}(\mathbb {F})\neq 2\) and \(|\mathbb {F}| \geq n^{2}+1\). A characterization of \(\phi ,\psi :{\varSigma }_{n} \rightarrow {\varSigma }_{n}\), for which at least one of them is surjective, satisfying

$ \det (\phi (x)+\psi (y))=\det (x+y)\qquad (x,y\in {\varSigma }_{n}) $

is given. Furthermore, if n is even and \(\phi ,\psi :Q_{n} \rightarrow Q_{n}\), for which ψ is surjective and ψ(0) = 0, satisfy

$\det (\phi (x)+\psi (y))=\det (x+y)\qquad (x,y\in Q_{n}), $

then ϕ = ψ and ψ must be a bijective linear map preserving the determinant.

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Acknowledgements

The authors would like to thank anonymous referee(s) for reviewing this manuscript.

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Correspondence to Kijti Rodtes.

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Sanguanwong, R., Rodtes, K. Determinants Preserving Maps on the Spaces of Symmetric Matrices and Skew-Symmetric Matrices. Vietnam J. Math. (2022). https://doi.org/10.1007/s10013-022-00569-0

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  • DOI: https://doi.org/10.1007/s10013-022-00569-0

Keywords

  • Determinant
  • Preserving problems
  • Symmetric matrix
  • Skew-symmetric matrix

Mathematics Subject Classification (2010)

  • 15A15