# Dieudonné theorem on linear preservers of the singular matrices

Let $\mathbb{F}$ be an arbitrary field. Consider $\mathcal{M}_{n}(\mathbb{F})$, the vector space of all $n\times n$ matrices over $\mathbb{F}$. Moreover, let $\mathcal{GL}_{n}(\mathbb{F})$ be the full linear group of nonsingular $n\times n$ matrices over $\mathbb{F}$.

###### Theorem 1.

For a linear automorphism $\varphi:\mathcal{M}_{n}(\mathbb{F})\longrightarrow\mathcal{M}_{n}(\mathbb{F})$ the following conditions are equivalent:
(i) $\displaystyle\forall\,A\in\mathcal{M}_{n}(\mathbb{F}):\,\det(A)=0\,\Rightarrow% \,\det(\varphi(A))=0$, (ii) either $\displaystyle\exists\,P,Q\in\mathcal{GL}_{n}(\mathbb{F})\,\forall\,A\in% \mathcal{M}_{n}(\mathbb{F}):\,\varphi(A)=PAQ$, or $\displaystyle\exists\,P,Q\in\mathcal{GL}_{n}(\mathbb{F})\,\forall\,A\in% \mathcal{M}_{n}(\mathbb{F}):\,\varphi(A)=PA^{\top}Q$.

The original proof [D] of the nontrivial implication (i) $\Rightarrow$ (ii) is based on the fundamental theorem of projective geometry.

## References

• D J. Dieudonné, Sur une généralisation du groupe orthogonal à quatre variables, Arch. Math. 1: 282–287 (1949).
Title Dieudonné theorem on linear preservers of the singular matrices DieudonneTheoremOnLinearPreserversOfTheSingularMatrices 2013-03-22 19:19:49 2013-03-22 19:19:49 kammerer (26336) kammerer (26336) 8 kammerer (26336) Theorem msc 15A15 msc 15A04 FundamentalTheoremOfProjectiveGeometry FrobeniusTheoremOnLinearDeterminantPreservers