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
Canonical name	DieudonneTheoremOnLinearPreserversOfTheSingularMatrices
Date of creation	2013-03-22 19:19:49
Last modified on	2013-03-22 19:19:49
Owner	kammerer (26336)
Last modified by	kammerer (26336)
Numerical id	8
Author	kammerer (26336)
Entry type	Theorem
Classification	msc 15A15
Classification	msc 15A04
Related topic	FundamentalTheoremOfProjectiveGeometry
Related topic	FrobeniusTheoremOnLinearDeterminantPreservers