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).

单词	FrobeniusTheoremOnLinearDeterminantPreservers
释义	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).
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