Dieudonné theorem on linear preservers of the singular matrices

Let 𝔽 be an arbitrary field. Consider n(𝔽), the vector spaceMathworldPlanetmath of all n×n matrices over 𝔽. Moreover, let 𝒢n(𝔽) be the full linear group of nonsingularPlanetmathPlanetmath n×n matrices over 𝔽.

Theorem 1.

For a linear automorphismMathworldPlanetmathPlanetmathPlanetmath φ:Mn(F)Mn(F) the following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:
(i) AMn(F):det(A)=0det(φ(A))=0, (ii) either P,QGLn(F)AMn(F):φ(A)=PAQ, or P,QGLn(F)AMn(F):φ(A)=PAQ.

The original proof [D] of the nontrivial implicationMathworldPlanetmath (i) (ii) is based on the fundamental theorem of projective geometryMathworldPlanetmath.


  • D J. Dieudonné, Sur une généralisation du groupe orthogonalMathworldPlanetmathPlanetmathPlanetmath 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