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).

单词	DieudonneTheoremOnLinearPreserversOfTheSingularMatrices
释义	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).
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