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单词 FrobeniusTheoremOnLinearDeterminantPreservers
释义

Frobenius theorem on linear determinant preservers


Let 𝔽 be an arbitrary field. Consider n(𝔽), the vector spaceMathworldPlanetmath of all n×n matrices over 𝔽. Let 𝒢n(𝔽) be the set of all nonsingular matrices Pn(𝔽).

Definition 1.

A linear endomorphismPlanetmathPlanetmath φ:Mn(F)Mn(F) is said to be in standard form, if either P,QGLn(F)AMn(F):φ(A)=PAQ or P,QGLn(F)AMn(F):φ(A)=PAQ.

The classical on linear preservers of the determinantMathworldPlanetmath function [GF] reads as follows.

Theorem 2.

If φ:Mn(C)Mn(C) is a linear automorphism such that det(φ(A))=det(A) for all AMn(C), then φ is in standard form with
det(PQ)=1.

It is well known that the can be strengthened.

Theorem 3.

Let F be an arbitrary field and let φ:Mn(F)Mn(F) be a linear endomorphism. Then the following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:
(i)det(φ(A))=det(A) for all AMn(F),(ii)φ is in standard form with det(PQ)=1.

The above strengthened version of the can be derived from the Dieudonné theorem on linear preservers of the singular matrices.

References

  • GF G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber., Preuss. Akad. Wiss., Berlin, 1897 (994–1015).
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更新时间:2025/5/4 20:24:26