# Frobenius theorem on linear determinant preservers

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

###### Definition 1.

A linear endomorphism $\varphi:\mathcal{M}_{n}(\mathbb{F})\longrightarrow\mathcal{M}_{n}(\mathbb{F})$ is said to be in standard form, if either $\exists\,P,Q\in\mathcal{GL}_{n}(\mathbb{F})\,\forall\,A\in\mathcal{M}_{n}(% \mathbb{F}):\,\varphi(A)=PAQ$ or $\exists\,P,Q\in\mathcal{GL}_{n}(\mathbb{F})\,\forall\,A\in\mathcal{M}_{n}(% \mathbb{F}):\,\varphi(A)=PA^{\top}Q$.

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

###### Theorem 2.

If $\varphi:\mathcal{M}_{n}(\mathbb{C})\longrightarrow\mathcal{M}_{n}(\mathbb{C})$ is a linear automorphism such that $\det(\varphi(A))=\det(A)$ for all $A\in\mathcal{M}_{n}(\mathbb{C})$, then $\varphi$ is in standard form with
$\det(PQ)=1$.

It is well known that the can be strengthened.

###### Theorem 3.

Let $\mathbb{F}$ be an arbitrary field and let $\varphi:\mathcal{M}_{n}(\mathbb{F})\longrightarrow\mathcal{M}_{n}(\mathbb{F})$ be a linear endomorphism. Then the following conditions are equivalent:
(i) $\det(\varphi(A))=\det(A)$ for all $A\in\mathcal{M}_{n}(\mathbb{F})$, (ii) $\varphi$ 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).
Title Frobenius theorem on linear determinant preservers FrobeniusTheoremOnLinearDeterminantPreservers 2013-03-22 19:19:52 2013-03-22 19:19:52 kammerer (26336) kammerer (26336) 7 kammerer (26336) Theorem msc 15A04 msc 15A15 DieudonneTheoremOnLinearPreserversOfTheSingularMatrices