Frobenius theorem on linear determinant preservers

Let $𝔽$ be an arbitrary field. Consider ${ℳ}_n(𝔽)$, the vector space of all $n\times n$ matrices over $𝔽$. Let $𝒢{ℒ}_n(𝔽)$ be the set of all nonsingular matrices $P\in {ℳ}_n(𝔽)$.

Definition 1.

A linear endomorphism $\phi \mathrm{:}{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}\mathrm{⟶}{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}$ is said to be in standard form, if either $\mathrm{\exists }P\mathrm{,}Q\mathrm{\in }\mathrm{G}\mathit{}{\mathrm{L}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}\mathit{}\mathrm{\forall }A\mathrm{\in }{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}\mathrm{:}\phi \mathit{}\mathrm{(}A\mathrm{)}\mathrm{=}P\mathit{}A\mathit{}Q$ or $\mathrm{\exists }P\mathrm{,}Q\mathrm{\in }\mathrm{G}\mathit{}{\mathrm{L}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}\mathit{}\mathrm{\forall }A\mathrm{\in }{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}\mathrm{:}\phi \mathit{}\mathrm{(}A\mathrm{)}\mathrm{=}P\mathit{}{A}^{\mathrm{\top }}\mathit{}Q$.

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

Theorem 2.

If $\phi \mathrm{:}{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}\mathrm{⟶}{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$ is a linear automorphism such that $\det \mathit{}\mathrm{(}\phi \mathit{}\mathrm{(}A\mathrm{)}\mathrm{)}\mathrm{=}\det \mathit{}\mathrm{(}A\mathrm{)}$ for all $A\mathrm{\in }{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$, then $\phi$ is in standard form with
$\det \mathit{}\mathrm{(}P\mathit{}Q\mathrm{)}\mathrm{=}\mathrm{1}$.

It is well known that the can be strengthened.

Theorem 3.

Let $\mathrm{F}$ be an arbitrary field and let $\phi \mathrm{:}{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}\mathrm{⟶}{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}$ be a linear endomorphism. Then the following conditions are equivalent:
(i) $\det \mathit{}\mathrm{(}\phi \mathit{}\mathrm{(}A\mathrm{)}\mathrm{)}\mathrm{=}\det \mathit{}\mathrm{(}A\mathrm{)}$ for all $A\mathrm{\in }{\mathrm{M}}_n\mathit{}\mathrm{(}\mathrm{F}\mathrm{)}$, (ii) $\phi$ is in standard form with $\det \mathit{}\mathrm{(}P\mathit{}Q\mathrm{)}\mathrm{=}\mathrm{1}$.

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

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