Frobenius theorem on linear determinant preservers
Let be an arbitrary field. Consider , the vector space of all matrices over . Let be the set of all nonsingular matrices .
Definition 1.
A linear endomorphism is said to be in standard form, if either or .
The classical on linear preservers of the determinant function [GF] reads as follows.
Theorem 2.
If is a linear automorphism such that for all , then is in standard form with
.
It is well known that the can be strengthened.
Theorem 3.
Let be an arbitrary field and let be a linear endomorphism. Then the following conditions are equivalent:
(i) for all ,(ii) is in standard form with .
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).