Frobenius theorem on linear determinant preservers

Let 𝔽 be an arbitrary field. Consider β„³n⁒(𝔽), the vector spaceMathworldPlanetmath of all nΓ—n matrices over 𝔽. Let 𝒒⁒ℒn⁒(𝔽) be the set of all nonsingular matrices Pβˆˆβ„³n⁒(𝔽).

Definition 1.

A linear endomorphismPlanetmathPlanetmath Ο†:Mn⁒(F)⟢Mn⁒(F) is said to be in standard form, if either βˆƒP,Q∈G⁒Ln⁒(F)β’βˆ€A∈Mn⁒(F):φ⁒(A)=P⁒A⁒Q or βˆƒP,Q∈G⁒Ln⁒(F)β’βˆ€A∈Mn⁒(F):φ⁒(A)=P⁒A⊀⁒Q.

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

Theorem 2.

If Ο†:Mn⁒(C)⟢Mn⁒(C) is a linear automorphism such that det⁑(φ⁒(A))=det⁑(A) for all A∈Mn⁒(C), then Ο† is in standard form with

It is well known that the can be strengthened.

Theorem 3.

Let F be an arbitrary field and let Ο†:Mn⁒(F)⟢Mn⁒(F) be a linear endomorphism. Then the following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:
(i) det⁑(φ⁒(A))=det⁑(A) for all A∈Mn⁒(F), (ii) Ο† is in standard form with det⁑(P⁒Q)=1.

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


  • GF G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber., Preuss. Akad. Wiss., Berlin, 1897 (994–1015).
Title Frobenius theorem on linear determinant preservers
Canonical name FrobeniusTheoremOnLinearDeterminantPreservers
Date of creation 2013-03-22 19:19:52
Last modified on 2013-03-22 19:19:52
Owner kammerer (26336)
Last modified by kammerer (26336)
Numerical id 7
Author kammerer (26336)
Entry type Theorem
Classification msc 15A04
Classification msc 15A15
Related topic DieudonneTheoremOnLinearPreserversOfTheSingularMatrices